Daily Papers

We verify that persistent observers in causally invariant hypergraph substrates satisfy the conditions of the Conant-Ashby Good Regulator Theorem. Building on Wolfram's hypergraph physics and Vanchurin's neural network cosmology, we formalize persistent observers as entities that minimize prediction error at their boundary with the environment. Applying a modern reformulation of the Conant-Ashby theorem, we demonstrate that hypergraph observers satisfy Good Regulator conditions, requiring them to maintain internal models. Once an internal model with loss function exists, the emergence of a Fisher information metric follows from standard information geometry. Invoking Amari's uniqueness theorem for reparameterization-invariant gradients, we show that natural gradient descent is the unique admissible learning rule. Under the ansatz M=F^2 for exponential family observers and one specific convergence time functional, we derive a closed-form formula for the regime parameter alpha in Vanchurin's Type II framework, with a quantum-classical threshold at kappa(F)=2. However, three alternative convergence models do not reproduce this result, so this prediction is strongly model-dependent. We further introduce the directional regime parameter alpha_{v_k} and the trace-free deviation tensor, showing that a single observer can simultaneously occupy different Vanchurin regimes along different eigendirections of the Fisher metric. This connects Wolfram and Vanchurin frameworks through established theorems, providing approximately 25-30% novel contribution.

We revisit the I/O complexity of attention in large language models. Given query-key-value matrices Q,K,VinR^{ntimes d}, and a machine with fast memory size M, the goal is to compute the "attention matrix" A=softmax(Q K ^{top}/d) V with the minimal number of data transfers between fast and slow memory. Existing methods in the literature, most notably FlashAttention and its variants, incur an I/O cost that depends quadratically on n, while a trivial lower bound only requires Ω(nd) I/O's to read the inputs and write the output. In this work, we present a technique for computing attention where the I/O cost only depends almost-linearly on n in most parameter regimes. This is achieved by developing I/O-efficient algorithms inspired by the recent approximate attention framework of Alman and Song. We also prove corresponding lower bounds in each parameter regime to show that our algorithms are indeed close to I/O-optimal.

Deep learning has recently revealed the existence of scaling laws, demonstrating that model performance follows predictable trends based on dataset and model sizes. Inspired by these findings and fascinating phenomena emerging in the over-parameterized regime, we examine a parallel direction: do similar scaling laws govern predictive uncertainties in deep learning? In identifiable parametric models, such scaling laws can be derived in a straightforward manner by treating model parameters in a Bayesian way. In this case, for example, we obtain O(1/N) contraction rates for epistemic uncertainty with respect to the number of data N. However, in over-parameterized models, these guarantees do not hold, leading to largely unexplored behaviors. In this work, we empirically show the existence of scaling laws associated with various measures of predictive uncertainty with respect to dataset and model sizes. Through experiments on vision and language tasks, we observe such scaling laws for in- and out-of-distribution predictive uncertainty estimated through popular approximate Bayesian inference and ensemble methods. Besides the elegance of scaling laws and the practical utility of extrapolating uncertainties to larger data or models, this work provides strong evidence to dispel recurring skepticism against Bayesian approaches: "In many applications of deep learning we have so much data available: what do we need Bayes for?". Our findings show that "so much data" is typically not enough to make epistemic uncertainty negligible.

It is believed that Gradient Descent (GD) induces an implicit bias towards good generalization in training machine learning models. This paper provides a fine-grained analysis of the dynamics of GD for the matrix sensing problem, whose goal is to recover a low-rank ground-truth matrix from near-isotropic linear measurements. It is shown that GD with small initialization behaves similarly to the greedy low-rank learning heuristics (Li et al., 2020) and follows an incremental learning procedure (Gissin et al., 2019): GD sequentially learns solutions with increasing ranks until it recovers the ground truth matrix. Compared to existing works which only analyze the first learning phase for rank-1 solutions, our result provides characterizations for the whole learning process. Moreover, besides the over-parameterized regime that many prior works focused on, our analysis of the incremental learning procedure also applies to the under-parameterized regime. Finally, we conduct numerical experiments to confirm our theoretical findings.

Second-order training methods have better convergence properties than gradient descent but are rarely used in practice for large-scale training due to their computational overhead. This can be viewed as a hardware limitation (imposed by digital computers). Here we show that natural gradient descent (NGD), a second-order method, can have a similar computational complexity per iteration to a first-order method, when employing appropriate hardware. We present a new hybrid digital-analog algorithm for training neural networks that is equivalent to NGD in a certain parameter regime but avoids prohibitively costly linear system solves. Our algorithm exploits the thermodynamic properties of an analog system at equilibrium, and hence requires an analog thermodynamic computer. The training occurs in a hybrid digital-analog loop, where the gradient and Fisher information matrix (or any other positive semi-definite curvature matrix) are calculated at given time intervals while the analog dynamics take place. We numerically demonstrate the superiority of this approach over state-of-the-art digital first- and second-order training methods on classification tasks and language model fine-tuning tasks.

Ensemble Kalman Inversion (EKI) is a derivative-free, ensemble-based method for inverse and optimization problems. Its continuous-time formulation can be interpreted as an interacting particle system driven by a Kalman-type preconditioned descent direction. A well-known limitation of this dynamics is the possible premature collapse of the covariance of the ensemble, which makes the method sensitive to the initial ensemble. We introduce a second-order particle system in which the particles evolve according to an inertial dynamics. The model combines a Kalman-type relaxation force with damping, attraction towards the ensemble mean, and a short-range repulsive interaction designed to counteract ensemble collapse. The resulting dynamics can be interpreted as a heavy-ball reformulation of continuous-time EKI enriched by competing attractive and repulsive mechanisms. For linear inverse problems, we analyze the induced mean and fluctuation dynamics and identify a parameter regime in which fully collapsed configurations are linearly unstable. We further characterize asymptotic equilibria through a constrained optimality condition on the subspace retained by the limiting ensemble covariance and derive an exponential decay estimate. Numerical experiments illustrate the effect of inertia and repulsion on the ensemble dynamics and compare the proposed second-order method with first-order EKI-type

Adversarial examples provoke weak reliability and potential security issues in deep neural networks. Although adversarial training has been widely studied to improve adversarial robustness, it works in an over-parameterized regime and requires high computations and large memory budgets. To bridge adversarial robustness and model compression, we propose a novel adversarial pruning method, Masking Adversarial Damage (MAD) that employs second-order information of adversarial loss. By using it, we can accurately estimate adversarial saliency for model parameters and determine which parameters can be pruned without weakening adversarial robustness. Furthermore, we reveal that model parameters of initial layer are highly sensitive to the adversarial examples and show that compressed feature representation retains semantic information for the target objects. Through extensive experiments on three public datasets, we demonstrate that MAD effectively prunes adversarially trained networks without loosing adversarial robustness and shows better performance than previous adversarial pruning methods.

Modern language models use a single matrix for input embedding and output projection. This couples two distinct objectives: token representation and discrimination over a vocabulary. This work introduces Leviathan, a Transformer architecture that replaces the input embedding matrix with learned embedding vectorization (LEV), a compact continuous mapping from token indices to embeddings. Leviathan's output head remains untied for a parameter increase of as low as 0.2%. Under controlled comparisons with identical Transformer backbones, Leviathan consistently improves language modeling performance over standard tied-embedding baselines across a 200M-1.2B parameter regime on The Pile with gains that grow during training. At 1.2B scale, Leviathan reduces validation perplexity by 9%, requires 2.1times fewer training tokens to reach the tied baseline's final loss, and improves on all six downstream benchmarks evaluated, including a 30% reduction in LAMBADA perplexity. Frequency-stratified analysis reveals gains to be concentrated in rare tokens, where continuous parameterization reduces perplexity by 81%, falling to near zero for the most frequent.

One of the fundamental challenges in enabling fault-tolerant quantum computation is realising fast enough quantum decoders. We present a new two-stage decoder that accelerates the decoding cycle and boosts accuracy. In the first stage, a partial decoder based on belief propagation is used to correct errors that occurred with high probability. In the second stage, a conventional decoder corrects any remaining errors. We study the performance of our two-stage decoder with simulations using the surface code under circuit-level noise. When the conventional decoder is minimum-weight perfect matching, adding the partial decoder decreases bandwidth requirements, increases speed and improves logical accuracy. Specifically, we observe partial decoding consistently speeds up the minimum-weight perfect matching stage by between 2x-4x on average depending on the parameter regime, and raises the threshold from 0.94% to 1.02%.

Temporal Difference (TD) algorithms are widely used in Deep Reinforcement Learning (RL). Their performance is heavily influenced by the size of the neural network. While in supervised learning, the regime of over-parameterization and its benefits are well understood, the situation in RL is much less clear. In this paper, we present a theoretical analysis of the influence of network size and l_2-regularization on performance. We identify the ratio between the number of parameters and the number of visited states as a crucial factor and define over-parameterization as the regime when it is larger than one. Furthermore, we observe a double descent phenomenon, i.e., a sudden drop in performance around the parameter/state ratio of one. Leveraging random features and the lazy training regime, we study the regularized Least-Square Temporal Difference (LSTD) algorithm in an asymptotic regime, as both the number of parameters and states go to infinity, maintaining a constant ratio. We derive deterministic limits of both the empirical and the true Mean-Squared Bellman Error (MSBE) that feature correction terms responsible for the double descent. Correction terms vanish when the l_2-regularization is increased or the number of unvisited states goes to zero. Numerical experiments with synthetic and small real-world environments closely match the theoretical predictions.

We study the parameter placement problem: given a fixed budget of k trainable entries within the B matrix of a LoRA adapter (A frozen), does the choice of which k matter? Under supervised fine-tuning, random and informed subsets achieve comparable performance. Under GRPO on base models, random placement fails to improve over the base model, while gradient-informed placement recovers standard LoRA accuracy. This regime dependence traces to gradient structure: SFT gradients are low-rank and directionally stable, so any subset accumulates coherent updates; GRPO gradients are high-rank and near-orthogonal across steps, so only elements with consistently signed gradients retain the learning signal. Our scoring procedure identifies these critical parameters in under 10 seconds at less than 0.5% of training cost. Selected parameters concentrate on residual-stream-writing projections (V, O, Down), stable across model families and scales (1.5B - 8B).

Recent work has highlighted the complex influence training hyperparameters, e.g., the number of training epochs, can have on the prunability of machine learning models. Perhaps surprisingly, a systematic approach to predict precisely how adjusting a specific hyperparameter will affect prunability remains elusive. To address this gap, we introduce a phenomenological model grounded in the statistical mechanics of learning. Our approach uses temperature-like and load-like parameters to model the impact of neural network (NN) training hyperparameters on pruning performance. A key empirical result we identify is a sharp transition phenomenon: depending on the value of a load-like parameter in the pruned model, increasing the value of a temperature-like parameter in the pre-pruned model may either enhance or impair subsequent pruning performance. Based on this transition, we build a three-regime model by taxonomizing the global structure of the pruned NN loss landscape. Our model reveals that the dichotomous effect of high temperature is associated with transitions between distinct types of global structures in the post-pruned model. Based on our results, we present three case-studies: 1) determining whether to increase or decrease a hyperparameter for improved pruning; 2) selecting the best model to prune from a family of models; and 3) tuning the hyperparameter of the Sharpness Aware Minimization method for better pruning performance.

Vision Transformers rely on positional embeddings and class tokens that encode fixed spatial priors. While effective for natural images, these priors may hinder generalization when spatial layout is weakly informative or inconsistent, a frequent condition in medical imaging and edge-deployed clinical systems. We introduce ZACH-ViT (Zero-token Adaptive Compact Hierarchical Vision Transformer), a compact Vision Transformer that removes both positional embeddings and the [CLS] token, achieving permutation invariance through global average pooling over patch representations. The term "Zero-token" specifically refers to removing the dedicated [CLS] aggregation token and positional embeddings; patch tokens remain unchanged and are processed normally. Adaptive residual projections preserve training stability in compact configurations while maintaining a strict parameter budget. Evaluation is performed across seven MedMNIST datasets spanning binary and multi-class tasks under a strict few-shot protocol (50 samples per class, fixed hyperparameters, five random seeds). The empirical analysis demonstrates regime-dependent behavior: ZACH-ViT (0.25M parameters, trained from scratch) achieves its strongest advantage on BloodMNIST and remains competitive with TransMIL on PathMNIST, while its relative advantage decreases on datasets with strong anatomical priors (OCTMNIST, OrganAMNIST), consistent with the architectural hypothesis. These findings support the view that aligning architectural inductive bias with data structure can be more important than pursuing universal benchmark dominance. Despite its minimal size and lack of pretraining, ZACH-ViT achieves competitive performance while maintaining sub-second inference times, supporting deployment in resource-constrained clinical environments. Code and models are available at https://github.com/Bluesman79/ZACH-ViT.

We obtain (small-parameter) well-posedness for the (space-time periodic) Phi^4 equation in the full subcritical regime in the context of regularity structures based on multi-indices. As opposed to Hairer's more extrinsic tree-based setting, due to the intrinsic description encoded by multi-indices, it is not possible to obtain a solution theory via the standard fixed-point argument. Instead, we develop a more intrinsic approach for existence using a variant of the continuity method from classical PDE theory based on a priori estimates for a new `robust' formulation of the equation. This formulation also allows us to obtain uniqueness of solutions and continuity of the solution map in the model norm even at the limit of vanishing regularisation scale. Since our proof relies on the structure of the nonlinearity in only a mild way, we expect the same ideas to be sufficient to treat a more general class of equations.

Low-rank adaptation (LoRA) has become the standard approach for parameter-efficient fine-tuning of large language models (LLM), but our theoretical understanding of LoRA has been limited. In this work, we theoretically analyze LoRA fine-tuning in the neural tangent kernel (NTK) regime with N data points, showing: (i) full fine-tuning (without LoRA) admits a low-rank solution of rank rlesssim N; (ii) using LoRA with rank rgtrsim N eliminates spurious local minima, allowing gradient descent to find the low-rank solutions; (iii) the low-rank solution found using LoRA generalizes well.

Training neural networks requires increasing amounts of memory. Parameter sharing can reduce memory and communication costs, but existing methods assume networks have many identical layers and utilize hand-crafted sharing strategies that fail to generalize. We introduce Neural Parameter Allocation Search (NPAS), a novel task where the goal is to train a neural network given an arbitrary, fixed parameter budget. NPAS covers both low-budget regimes, which produce compact networks, as well as a novel high-budget regime, where additional capacity can be added to boost performance without increasing inference FLOPs. To address NPAS, we introduce Shapeshifter Networks (SSNs), which automatically learn where and how to share parameters in a network to support any parameter budget without requiring any changes to the architecture or loss function. NPAS and SSNs provide a complete framework for addressing generalized parameter sharing, and can also be combined with prior work for additional performance gains. We demonstrate the effectiveness of our approach using nine network architectures across four diverse tasks, including ImageNet classification and transformers.

Recent frontier large language models predominantly rely on Mixture-of-Experts (MoE) architectures. Despite empirical progress, there is still no principled understanding of how hyperparameters should scale with network width N, expert width N_e, number of experts M, sparsity K, and depth L to ensure both stability and optimal performance at scale. We take a principled step toward resolving this gap by analyzing three different scaling regimes: (I) co-scaling Nasymp N_e, (II) co-scaling Nasymp Masymp K, and (III) full proportional scaling of N, N_e, M, and K. For each regime, we develop a novel Dynamical Mean Field Theory (DMFT) description of the limiting training dynamics of MoEs that provides a formal foundation for our analysis. Within this framework, we derive the unique parameterization for SGD and Adam satisfying all maximal-update (μ) desiderata. We then show that the resulting μP prescription does not reliably induce monotonic improvement with scale or robust learning-rate transfer. We trace these pathologies to scale-dependent observables in the aggregation dynamics, which motivates a refined set of desiderata that we term maximal scale stability. Guided by this principle, we derive a Maximally Scale-Stable Parameterization (MSSP) for both SGD and Adam in all three scaling regimes, and characterize the corresponding limiting dynamics - qualitatively distinct from the μP limit - through a separate DMFT analysis. Experiments verify that MSSP robustly recovers learning rate transfer and monotonic improvement with scale across regimes. Combined with existing depth-scaling theory, these results provide a complete scaling prescription for MoE architectures as a function of width, depth, expert width, and number of experts.

Direct Preference Optimization (DPO) is widely used after supervised fine-tuning (SFT) to align language models, yet empirical behavior under small backbones and modest data is under-specified. We systematically compare SFT-only, DPO-only, and staged SFT-to-DPO training alongside full fine-tuning (FFT) versus LoRA on a GPT-2-scale decoder, evaluating paraphrase detection and Shakespearean sonnet continuation. DPO yields small, task-dependent gains over strong SFT and can match competitive SFT accuracy without a warm start when the preference construction closely parallels the supervised objective. In contrast, parameterization dominates: FFT consistently outperforms LoRA at matched training depth, and LoRA does not reduce wall-clock time on our hardware. These findings indicate that, in this small-scale regime, supervised full-parameter adaptation remains the primary performance lever, while preference optimization and low-rank adaptation provide limited marginal returns.

Large language models (LLMs) and vision language models (VLMs) demonstrate excellent performance on a wide range of tasks by scaling up parameter counts from O(10^9) to O(10^{12}) levels and further beyond. These large scales make it impossible to adapt and deploy fully specialized models given a task of interest. Parameter-efficient fine-tuning (PEFT) emerges as a promising direction to tackle the adaptation and serving challenges for such large models. We categorize PEFT techniques into two types: intrusive and non-intrusive. Intrusive PEFT techniques directly change a model's internal architecture. Though more flexible, they introduce significant complexities for training and serving. Non-intrusive PEFT techniques leave the internal architecture unchanged and only adapt model-external parameters, such as embeddings for input. In this work, we describe AdaLink as a non-intrusive PEFT technique that achieves competitive performance compared to SoTA intrusive PEFT (LoRA) and full model fine-tuning (FT) on various tasks. We evaluate using both text-only and multimodal tasks, with experiments that account for both parameter-count scaling and training regime (with and without instruction tuning).

Long-horizon search agents accumulate large amounts of retrieved content across many tool calls, making context-budget efficiency increasingly important. A minimal intervention is to mask stale observations from the context as the trajectory progresses, but it remains unclear when this form of context management helps and why. We study observation masking through a systematic sweep over various agent backbones (4B to 284B parameters) and three retrievers on offline and live-web agentic search benchmarks. We find that the accuracy gain from masking follows an asymmetric inverted-U shape when plotted against the model's accuracy without context management: a plateau under weak retrievers, a peak when a strong retriever meets a mid-capacity model, and a sharp collapse when the model is saturated. This pattern reflects the interaction between retriever recall and the model's implicit filtering capacity, rather than either factor in isolation. Mechanistically, masking implements a token-for-turn trade-off: it removes observations the model has largely stopped attending to and pages the agent rarely re-opens. The added turns help when they convert failures into successes, but they fail when masking removes evidence the model would otherwise have used. We therefore reframe context management as a regime-dependent intervention and provide a holistic perspective for analyzing context use in agentic deep search. We release our scaffold and trajectories here (https://github.com/i-DeepSearch/observation-masking) to support future research.

We introduce Leap+Verify, a framework that applies speculative execution -- predicting future model weights and validating predictions before acceptance -- to accelerate neural network training. Inspired by speculative decoding in language model inference and by the Automatically Scalable Computation (ASC) architecture for program execution, Leap+Verify decomposes training into three dynamically detected regimes (chaotic, transition, stable) using activation-space cosine similarity as a real-time Lyapunov proxy signal. Within each regime, analytic weight predictors (momentum, linear, quadratic extrapolation) attempt to forecast model parameters K training steps ahead; predictions are accepted only when validated against a held-out loss criterion. We evaluate Leap+Verify on GPT-2 124M and Qwen 2.5-1.5B trained on WikiText-103 across five random seeds, sweeping prediction depth K in {5, 10, 25, 50, 75, 100}. Momentum-based prediction (Adam moment extrapolation) fails catastrophically at both scales, with predicted losses exceeding actuals by 100-10,000x -- a universal norm explosion in optimizer-state extrapolation. Finite-difference predictors (linear, quadratic) succeed where momentum fails: at 124M, they achieve 24% strict acceptance at K=5 in stable regimes; at 1.5B, they achieve 37% strict acceptance in transition regimes. The scale-dependent finding is in regime distribution: GPT-2 124M spends 34% of training in stable regime, while Qwen 1.5B spends 64% in chaotic regime and reaches stable in only 0-2 of 40 checkpoints. Larger models are more predictable when predictable, but less often predictable -- the practical bottleneck shifts from predictor accuracy to regime availability. Cross-seed results are highly consistent (less than 1% validation loss variance), and the three-regime framework produces identical phase boundaries (plus or minus 50 steps) across seeds.

Machine-learning-based parameterizations (i.e. representation of sub-grid processes) of global climate models or turbulent simulations have recently been proposed as a powerful alternative to physical, but empirical, representations, offering a lower computational cost and higher accuracy. Yet, those approaches still suffer from a lack of generalization and extrapolation beyond the training data, which is however critical to projecting climate change or unobserved regimes of turbulence. Here we show that a multi-fidelity approach, which integrates datasets of different accuracy and abundance, can provide the best of both worlds: the capacity to extrapolate leveraging the physically-based parameterization and a higher accuracy using the machine-learning-based parameterizations. In an application to climate modeling, the multi-fidelity framework yields more accurate climate projections without requiring major increase in computational resources. Our multi-fidelity randomized prior networks (MF-RPNs) combine physical parameterization data as low-fidelity and storm-resolving historical run's data as high-fidelity. To extrapolate beyond the training data, the MF-RPNs are tested on high-fidelity warming scenarios, +4K, data. We show the MF-RPN's capacity to return much more skillful predictions compared to either low- or high-fidelity (historical data) simulations trained only on one regime while providing trustworthy uncertainty quantification across a wide range of scenarios. Our approach paves the way for the use of machine-learning based methods that can optimally leverage historical observations or high-fidelity simulations and extrapolate to unseen regimes such as climate change.

Dynamic Tanh (DyT) removes LayerNorm by bounding activations with a learned tanh(alpha x). We show that this bounding is a regime-dependent implicit regularizer, not a uniformly beneficial replacement. Across GPT-2-family models spanning 64M to 3.78B parameters and 1M to 118M tokens, with Llama and ViT cross-checks, DyT improves validation loss by 27.3% at 64M/1M but worsens it by 18.8% at 64M/118M; the 1M benefit vanishes with capacity (+1.7% at 3.78B), while the 118M penalty reaches +27.9%. The mechanism is measurable: 49% of DyT activations saturate at 1M versus 23% at 118M, and a 500-step saturation heuristic classifies DyT's sign with 75% raw in-sample accuracy on the 12-cell GPT-2 calibration set (AUC 0.75; 64% when adding Scale 5 stress cells), correctly labels 3/3 Llama checks, but only reaches 50% raw leave-one-scale-out accuracy. Three interventions support the bounding explanation: HardTanh reproduces the regime pattern, increasing alpha at 118M monotonically reduces DyT's penalty, and vanilla+dropout(p=0.5) matches DyT's data-rich loss. We also localize Llama-DyT collapse to SwiGLU gating, where saturation separates collapse from convergence in a 3-seed component ablation (r=0.94). Scope: all experiments are compute-limited (T/P < 1.84), below Chinchilla-optimal training.

The Mixture of Experts (MoE) framework has become a popular architecture for large language models due to its superior performance over dense models. However, training MoEs from scratch in a large-scale regime is prohibitively expensive. Existing methods mitigate this by pre-training multiple dense expert models independently and using them to initialize an MoE. This is done by using experts' feed-forward network (FFN) to initialize the MoE's experts while merging other parameters. However, this method limits the reuse of dense model parameters to only the FFN layers, thereby constraining the advantages when "upcycling" these models into MoEs. We propose BAM (Branch-Attend-Mix), a simple yet effective method that addresses this shortcoming. BAM makes full use of specialized dense models by not only using their FFN to initialize the MoE layers but also leveraging experts' attention parameters fully by initializing them into a soft-variant of Mixture of Attention (MoA) layers. We explore two methods for upcycling attention parameters: 1) initializing separate attention experts from dense models including all attention parameters for the best model performance; and 2) sharing key and value parameters across all experts to facilitate for better inference efficiency. To further improve efficiency, we adopt a parallel attention transformer architecture to MoEs, which allows the attention experts and FFN experts to be computed concurrently. Our experiments on seed models ranging from 590 million to 2 billion parameters demonstrate that BAM surpasses baselines in both perplexity and downstream task performance, within the same computational and data constraints.

In this paper we consider a particular class of solutions of the Rayleigh-Boltzmann equation, known in the nonlinear setting as homoenergetic solutions, which have the form gleft( x,v,t right) =fleft( v-Lleft( tright)x,tright) where the matrix L(t) describes a shear flow deformation. We began this analysis in [22] where we rigorously proved the existence of a stationary non-equilibrium solution and established the different behaviour of the solutions for small and large values of the shear parameter, for cut-off collision kernels with homogeneity parameter 0leq gamma <1, including Maxwell molecules and hard potentials. In this paper, we concentrate in the case where the deformation term dominates the collision term for large times (hyperbolic-dominated regime). This occurs for collision kernels with gamma < 0 and in particular we focus on gamma in (-1,0). In such a hyperbolic-dominated regime, it appears challenging to provide a clear description of the long-term asymptotics of the solutions. Here we present a formal analysis of the long-time asymptotics for the distribution of velocities and provide the explicit form for the asymptotic profile. Additionally, we discuss the different asymptotic behaviour expected in the case of homogeneity gamma < -1. Furthermore, we provide a probabilistic interpretation describing a stochastic process consisting in a combination of collisions and shear flows. The tagged particle velocity {v(t)}_{tgeq 0} is a Markov process that arises from the combination of free flights in a shear flow along with random jumps caused by collisions.

Despite the widespread empirical success of ResNet, the generalization properties of deep ResNet are rarely explored beyond the lazy training regime. In this work, we investigate scaled ResNet in the limit of infinitely deep and wide neural networks, of which the gradient flow is described by a partial differential equation in the large-neural network limit, i.e., the mean-field regime. To derive the generalization bounds under this setting, our analysis necessitates a shift from the conventional time-invariant Gram matrix employed in the lazy training regime to a time-variant, distribution-dependent version. To this end, we provide a global lower bound on the minimum eigenvalue of the Gram matrix under the mean-field regime. Besides, for the traceability of the dynamic of Kullback-Leibler (KL) divergence, we establish the linear convergence of the empirical error and estimate the upper bound of the KL divergence over parameters distribution. Finally, we build the uniform convergence for generalization bound via Rademacher complexity. Our results offer new insights into the generalization ability of deep ResNet beyond the lazy training regime and contribute to advancing the understanding of the fundamental properties of deep neural networks.

Parameter-efficient fine-tuning (PEFT) is a standard way to adapt multimodal large language models, yet extremely low-rank settings -- especially rank-1 LoRA -- are often unstable. We show that this instability is not solely due to limited capacity: in the rank-1 regime, optimization is highly sensitive to the update direction. Concretely, pretrained vision and text features form mismatched anisotropic regions, yielding a dominant "gap" direction that acts like a translation component and disproportionately steers early gradients under rank-1 constraints. Analyzing pretrained representations, we identify a modality-gap axis that dominates early gradient flow, while a random rank-1 initialization is unlikely to align with it, leading to weak gradients and training collapse. We propose Gap-Init, a geometry-aware initialization that aligns the rank-1 LoRA direction with an estimated modality-gap vector from a small calibration set, while keeping the initial LoRA update zero. Across multiple vision-language tasks and backbones, Gap-Init consistently stabilizes rank-1 training and can match or outperform strong rank-8 baselines. Our results suggest that at the extreme low-rank limit, initial alignment can matter as much as rank itself.

Scaling limits, such as infinite-width limits, serve as promising theoretical tools to study large-scale models. However, it is widely believed that existing infinite-width theory does not faithfully explain the behavior of practical networks, especially those trained in standard parameterization (SP) meaning He initialization with a global learning rate. For instance, existing theory for SP predicts instability at large learning rates and vanishing feature learning at stable ones. In practice, however, optimal learning rates decay slower than theoretically predicted and networks exhibit both stable training and non-trivial feature learning, even at very large widths. Here, we show that this discrepancy is not fully explained by finite-width phenomena. Instead, we find a resolution through a finer-grained analysis of the regime previously considered unstable and therefore uninteresting. In particular, we show that, under cross-entropy (CE) loss, the unstable regime comprises two distinct sub-regimes: a catastrophically unstable regime and a more benign controlled divergence regime, where logits diverge but gradients and activations remain stable. Moreover, under large learning rates at the edge of the controlled divergence regime, there exists a well-defined infinite width limit where features continue to evolve in all the hidden layers. In experiments across optimizers, architectures, and data modalities, we validate that neural networks operate in this controlled divergence regime under CE loss but not under MSE loss. Our empirical evidence suggests that width-scaling considerations are surprisingly useful for predicting empirically maximal stable learning rate exponents which provide useful guidance on optimal learning rate exponents. Finally, our analysis clarifies the effectiveness and limitations of recently proposed layerwise learning rate scaling for standard initialization.

This paper focuses on over-parameterized deep neural networks (DNNs) with ReLU activation functions and proves that when the data distribution is well-separated, DNNs can achieve Bayes-optimal test error for classification while obtaining (nearly) zero-training error under the lazy training regime. For this purpose, we unify three interrelated concepts of overparameterization, benign overfitting, and the Lipschitz constant of DNNs. Our results indicate that interpolating with smoother functions leads to better generalization. Furthermore, we investigate the special case where interpolating smooth ground-truth functions is performed by DNNs under the Neural Tangent Kernel (NTK) regime for generalization. Our result demonstrates that the generalization error converges to a constant order that only depends on label noise and initialization noise, which theoretically verifies benign overfitting. Our analysis provides a tight lower bound on the normalized margin under non-smooth activation functions, as well as the minimum eigenvalue of NTK under high-dimensional settings, which has its own interest in learning theory.

On-policy distillation (OPD) is increasingly used to improve large language model reasoning, but its training dynamics remain poorly understood. We characterize the trajectory of OPD updates in parameter space and compare it with supervised fine-tuning (SFT) and reinforcement learning with verifiable rewards (RLVR). A suite of parameter-space diagnostics consistently places OPD in a relaxed off-principal regime: compared with SFT, its updates affect fewer weights and avoid principal directions more strongly, while compared with RLVR, they remain less tightly constrained. Beyond this static localization, OPD exhibits subspace locking: its cumulative updates rapidly enter a narrow low-dimensional channel. Constraining training to the update subspace formed early in training preserves OPD performance but substantially degrades SFT, indicating that the locked subspace is functionally sufficient for OPD. Control experiments further show that sparsifying the update tokens and shifting rollout generation off-policy preserve the rank dynamics, whereas mixing the OPD objective with RLVR changes them. Overall, these results suggest that OPD is not merely an intermediate point between SFT and RLVR, but induces its own update geometry in parameter space.

Strong collective interaction of atoms with an optical cavity causes normal mode splitting of the cavity's resonances, whose width is given by the collective coupling strength. At low optical density of the atomic cloud the intensity distribution of light in the cavity is ruled by the cavity's mode function, which is solely determined by its geometry. In this regime the dynamics of the coupled atom-cavity system is conveniently described by the open Dicke model, which we apply to calculating normal mode splitting generated by periodically ordered clouds in linear and ring cavities. We also show how to use normal mode splitting as witness for Wannier-Bloch oscillations in the tight-binding limit. At high optical density the atomic distribution contributes to shaping the mode function. This regime escapes the open Dicke model, but can be treated by a transfer matrix model provided the saturation parameter is low. Applying this latter model to an atomic cloud periodically ordered into a one-dimensional lattice, we observe the formation of photonic bands gaps competing with the normal mode splitting. We discuss the limitations of both models and point out possible pathways to generalized theories.

We perform an average case analysis of the generalization dynamics of large neural networks trained using gradient descent. We study the practically-relevant "high-dimensional" regime where the number of free parameters in the network is on the order of or even larger than the number of examples in the dataset. Using random matrix theory and exact solutions in linear models, we derive the generalization error and training error dynamics of learning and analyze how they depend on the dimensionality of data and signal to noise ratio of the learning problem. We find that the dynamics of gradient descent learning naturally protect against overtraining and overfitting in large networks. Overtraining is worst at intermediate network sizes, when the effective number of free parameters equals the number of samples, and thus can be reduced by making a network smaller or larger. Additionally, in the high-dimensional regime, low generalization error requires starting with small initial weights. We then turn to non-linear neural networks, and show that making networks very large does not harm their generalization performance. On the contrary, it can in fact reduce overtraining, even without early stopping or regularization of any sort. We identify two novel phenomena underlying this behavior in overcomplete models: first, there is a frozen subspace of the weights in which no learning occurs under gradient descent; and second, the statistical properties of the high-dimensional regime yield better-conditioned input correlations which protect against overtraining. We demonstrate that naive application of worst-case theories such as Rademacher complexity are inaccurate in predicting the generalization performance of deep neural networks, and derive an alternative bound which incorporates the frozen subspace and conditioning effects and qualitatively matches the behavior observed in simulation.

Generative foundation models are increasingly scaled in both width and depth, posing significant challenges for stable feature learning and reliable hyperparameter (HP) transfer across model sizes. While maximal update parameterization (μP) has provided a principled solution to both problems for width scaling, existing extensions to the joint width-depth scaling regime remain fragmented, architecture- and optimizer-specific, and often rely on technically involved theories. In this work, we develop a simple and unified spectral framework for μP under joint width-depth scaling. Considering residual networks of varying block depths, we first introduce a spectral μP condition that precisely characterizes how the norms of weights and their per-step updates should scale with width and depth, unifying previously disparate μP formulations as special cases. Building on this condition, we then derive a general recipe for implementing μP across a broad class of optimizers by mapping the spectral constraints to concrete HP parameterizations. This approach not only recovers existing μP formulations (e.g., for SGD and AdamW) but also naturally extends to a wider range of optimizers. Finally, experiments on GPT-2 style language models demonstrate that the proposed spectral μP condition preserves stable feature learning and enables robust HP transfer under width-depth scaling.

Pretrained language models (LMs) have shown compelling performance on various downstream tasks, but unfortunately they require a tremendous amount of inference compute. Knowledge distillation finds a path to compress LMs to small ones with a teacher-student paradigm. However, when the capacity gap between the teacher and the student is large, a curse of capacity gap appears, invoking a deficiency in distilling LMs. While a few studies have been carried out to fill the gap, the curse is not yet well tackled. In this paper, we aim at lifting the curse of capacity gap via enlarging the capacity of the student without notably increasing the inference compute. Largely motivated by sparse activation regime of mixture of experts (MoE), we propose a mixture of minimal experts (MiniMoE), which imposes extra parameters to the student but introduces almost no additional inference compute. Experimental results on GLUE and CoNLL demonstrate the curse of capacity gap is lifted by the magic of MiniMoE to a large extent. MiniMoE also achieves the state-of-the-art performance at small FLOPs compared with a range of competitive baselines. With a compression rate as much as sim50times, MiniMoE preserves sim95\% GLUE score of the teacher.

We present LongLoRA, an efficient fine-tuning approach that extends the context sizes of pre-trained large language models (LLMs), with limited computation cost. Typically, training LLMs with long context sizes is computationally expensive, requiring extensive training hours and GPU resources. For example, training on the context length of 8192 needs 16x computational costs in self-attention layers as that of 2048. In this paper, we speed up the context extension of LLMs in two aspects. On the one hand, although dense global attention is needed during inference, fine-tuning the model can be effectively and efficiently done by sparse local attention. The proposed shift short attention effectively enables context extension, leading to non-trivial computation saving with similar performance to fine-tuning with vanilla attention. Particularly, it can be implemented with only two lines of code in training, while being optional in inference. On the other hand, we revisit the parameter-efficient fine-tuning regime for context expansion. Notably, we find that LoRA for context extension works well under the premise of trainable embedding and normalization. LongLoRA demonstrates strong empirical results on various tasks on LLaMA2 models from 7B/13B to 70B. LongLoRA adopts LLaMA2 7B from 4k context to 100k, or LLaMA2 70B to 32k on a single 8x A100 machine. LongLoRA extends models' context while retaining their original architectures, and is compatible with most existing techniques, like FlashAttention-2. In addition, to make LongLoRA practical, we collect a dataset, LongQA, for supervised fine-tuning. It contains more than 3k long context question-answer pairs.

While large language models (LLMs) often adopt finetuning to unlock their capabilities for downstream applications, our understanding on the inductive biases (especially the scaling properties) of different finetuning methods is still limited. To fill this gap, we conduct systematic experiments studying whether and how different scaling factors, including LLM model size, pretraining data size, new finetuning parameter size and finetuning data size, affect the finetuning performance. We consider two types of finetuning -- full-model tuning (FMT) and parameter efficient tuning (PET, including prompt tuning and LoRA), and explore their scaling behaviors in the data-limited regime where the LLM model size substantially outweighs the finetuning data size. Based on two sets of pretrained bilingual LLMs from 1B to 16B and experiments on bilingual machine translation and multilingual summarization benchmarks, we find that 1) LLM finetuning follows a powerbased multiplicative joint scaling law between finetuning data size and each other scaling factor; 2) LLM finetuning benefits more from LLM model scaling than pretraining data scaling, and PET parameter scaling is generally ineffective; and 3) the optimal finetuning method is highly task- and finetuning data-dependent. We hope our findings could shed light on understanding, selecting and developing LLM finetuning methods.

The Muon optimizer has demonstrated promising performance in pre-training large language models through gradient (or momentum) orthogonalization. In this work, we propose a simple yet effective enhancement to Muon, namely Muon+, which introduces an additional normalization step after orthogonalization. We demonstrate the effectiveness of Muon+ through extensive pre-training experiments across a wide range of model scales and architectures. Our evaluation includes GPT-style models ranging from 130M to 774M parameters and LLaMA-style models ranging from 60M to 1B parameters. We comprehensively evaluate the effectiveness of Muon+ in the compute-optimal training regime and further extend the token-to-parameter (T2P) ratio to an industrial level of approx 200. Experimental results show that Muon+ provides a consistent boost on training and validation perplexity over Muon. We provide our code here: https://github.com/K1seki221/MuonPlus.

Despite significant work on low-bit quantization-aware training (QAT), there is still a large accuracy gap between such techniques and native training. To address this, we introduce CAGE (Curvature-Aware Gradient Estimation), a new QAT method that augments the straight-through estimator (STE) gradient with a curvature-aware correction designed to counteract the loss increase induced by quantization. CAGE is derived from a multi-objective view of QAT that balances loss minimization with adherence to quantization constraints, yielding a principled correction term that depends on local curvature information. On the theoretical side, we introduce the notion of Pareto-optimal solutions for quantized optimization, and establish that CAGE yields strong convergence guarantees in the smooth non-convex setting. In terms of implementation, our approach is optimizer-agnostic, but we provide a highly-efficient implementation that leverages Adam statistics. When pre-training Llama-style models of up to 800M-parameters, CAGE recovers over 10% of the quantization-induced loss increase in the W4A4 regime over outlier-mitigation methods. These results indicate that curvature-aware gradient corrections can bridge the remaining performance gap beyond current outlier-handling methods.

Generating synthetic data through generative models is gaining interest in the ML community and beyond, promising a future where datasets can be tailored to individual needs. Unfortunately, synthetic data is usually not perfect, resulting in potential errors in downstream tasks. In this work we explore how the generative process affects the downstream ML task. We show that the naive synthetic data approach -- using synthetic data as if it is real -- leads to downstream models and analyses that do not generalize well to real data. As a first step towards better ML in the synthetic data regime, we introduce Deep Generative Ensemble (DGE) -- a framework inspired by Deep Ensembles that aims to implicitly approximate the posterior distribution over the generative process model parameters. DGE improves downstream model training, evaluation, and uncertainty quantification, vastly outperforming the naive approach on average. The largest improvements are achieved for minority classes and low-density regions of the original data, for which the generative uncertainty is largest.

Data arrives at our senses as a continuous stream, smoothly transforming from one instant to the next. These smooth transformations can be viewed as continuous symmetries of the environment that we inhabit, defining equivalence relations between stimuli over time. In machine learning, neural network architectures that respect symmetries of their data are called equivariant and have provable benefits in terms of generalization ability and sample efficiency. To date, however, equivariance has been considered only for static transformations and feed-forward networks, limiting its applicability to sequence models, such as recurrent neural networks (RNNs), and corresponding time-parameterized sequence transformations. In this work, we extend equivariant network theory to this regime of `flows' -- one-parameter Lie subgroups capturing natural transformations over time, such as visual motion. We begin by showing that standard RNNs are generally not flow equivariant: their hidden states fail to transform in a geometrically structured manner for moving stimuli. We then show how flow equivariance can be introduced, and demonstrate that these models significantly outperform their non-equivariant counterparts in terms of training speed, length generalization, and velocity generalization, on both next step prediction and sequence classification. We present this work as a first step towards building sequence models that respect the time-parameterized symmetries which govern the world around us.

Reinforcement Learning with Verifiable Rewards (RLVR) reliably improves the reasoning performance of large language models, yet it appears to modify only a small fraction of parameters. We revisit this paradox and show that sparsity is a surface artifact of a model-conditioned optimization bias: for a fixed pretrained model, updates consistently localize to preferred parameter regions, highly consistent across runs and largely invariant to datasets and RL recipes. We mechanistically explain these dynamics with a Three-Gate Theory: Gate I (KL Anchor) imposes a KL-constrained update; Gate II (Model Geometry) steers the step off principal directions into low-curvature, spectrum-preserving subspaces; and Gate III (Precision) hides micro-updates in non-preferred regions, making the off-principal bias appear as sparsity. We then validate this theory and, for the first time, provide a parameter-level characterization of RLVR's learning dynamics: RLVR learns off principal directions in weight space, achieving gains via minimal spectral drift, reduced principal-subspace rotation, and off-principal update alignment. In contrast, SFT targets principal weights, distorts the spectrum, and even lags RLVR. Together, these results provide the first parameter-space account of RLVR's training dynamics, revealing clear regularities in how parameters evolve. Crucially, we show that RL operates in a distinct optimization regime from SFT, so directly adapting SFT-era parameter-efficient fine-tuning (PEFT) methods can be flawed, as evidenced by our case studies on advanced sparse fine-tuning and LoRA variants. We hope this work charts a path toward a white-box understanding of RLVR and the design of geometry-aware, RLVR-native learning algorithms, rather than repurposed SFT-era heuristics.

The search for life beyond Earth will depend on detecting faint signatures in the atmospheres of potentially habitable exoplanets. Interpreting those signatures requires understanding the host planet's climate: the same molecule may signal life on one planet and abiotic chemistry on another. Global climate models (GCMs) provide this understanding, but individual runs can require up to millions of core-hours and substantial domain expert time. Machine-learning emulators could remove this bottleneck, but progress has been limited by the absence of a curated, multi-model exoclimate dataset. We introduce ThousandWorlds, an ML-ready benchmark for exoclimate emulation and for the broader regime of low-data, multi-simulator, parameter-to-field regression. The dataset contains approximately 1800 simulations from five GCMs, mapping eight planet parameters to 3D atmospheric fields including temperature, humidity, winds, clouds, and radiation. Three nested subsets define progressively harder challenges: single-simulator regression, multi-simulator regression with complete observations, and multi-simulator regression with structured missingness. We propose two evaluation protocols: one for ranking methods, and one that measures performance relative to the disagreement between GCMs themselves. We evaluate seven baselines spanning simple methods, deep learning, and Gaussian processes. GP-based methods perform best, suggesting that ThousandWorlds exposes a regime where off-the-shelf deep learning does not yet succeed. Data: https://doi.org/10.57967/hf/8695. Code: https://github.com/edstevenson/ThousandWorlds.

We present Recurrent Video Masked-Autoencoders (RVM): a novel approach to video representation learning that leverages recurrent computation to model the temporal structure of video data. RVM couples an asymmetric masking objective with a transformer-based recurrent neural network to aggregate information over time, training solely on a simple pixel reconstruction loss. This design yields a highly efficient "generalist" encoder: RVM achieves competitive performance with state-of-the-art video models (e.g. VideoMAE, V-JEPA) on video-level tasks like action classification, and point and object tracking, while matching or exceeding the performance of image models (e.g. DINOv2) on tasks that require strong geometric and dense spatial features. Notably, RVM achieves strong performance in the small-model regime without requiring knowledge distillation, exhibiting up to 30x greater parameter efficiency than competing video masked autoencoders. Finally, we demonstrate that RVM's recurrent nature allows for stable feature propagation over long temporal horizons with linear computational cost, overcoming some of the limitations of standard spatio-temporal attention-based video models. Ablation studies further highlight the factors driving the model's success, with qualitative results showing that RVM learns rich representations of scene semantics, structure, and motion.

The integration of large language models (LLMs) with vision-language (VL) tasks has been a transformative development in the realm of artificial intelligence, highlighting the potential of LLMs as a versatile general-purpose chatbot. However, the current trend in this evolution focuses on the integration of vision and language to create models that can operate in more diverse and real-world contexts. We present a novel approach, termed Bottleneck Adapter, specifically crafted for enhancing the multimodal functionalities of these complex models, enabling joint optimization of the entire multimodal LLM framework through a process known as Multimodal Model Tuning (MMT). Our approach utilizes lightweight adapters to connect the image encoder and LLM without the need for large, complex neural networks. Unlike the conventional modular training schemes, our approach adopts an end-to-end optimization regime, which, when combined with the adapters, facilitates the joint optimization using a significantly smaller parameter set. Our method exhibits robust performance with 90.12\% accuracy, outperforming both human-level performance (88.4\%) and LaVIN-7B (89.41\%).

Self-distillation (SD) is the process of first training a teacher model and then using its predictions to train a student model with the same architecture. Specifically, the student's objective function is big(xi*ell(teacher's predictions, student's predictions) + (1-xi)*ell(given labels, student's predictions)big), where ell is some loss function and xi is some parameter in [0,1]. Empirically, SD has been observed to provide performance gains in several settings. In this paper, we theoretically characterize the effect of SD in two supervised learning problems with noisy labels. We first analyze SD for regularized linear regression and show that in the high label noise regime, the optimal value of xi that minimizes the expected error in estimating the ground truth parameter is surprisingly greater than 1. Empirically, we show that xi > 1 works better than xi leq 1 even with the cross-entropy loss for several classification datasets when 50\% or 30\% of the labels are corrupted. Further, we quantify when optimal SD is better than optimal regularization. Next, we analyze SD in the case of logistic regression for binary classification with random label corruption and quantify the range of label corruption in which the student outperforms the teacher in terms of accuracy. To our knowledge, this is the first result of its kind for the cross-entropy loss.

Few-step video generation has been significantly advanced by consistency distillation. However, the performance of consistency-distilled models often degrades as more sampling steps are allocated at test time, limiting their effectiveness for any-step video diffusion. This limitation arises because consistency distillation replaces the original probability-flow ODE trajectory with a consistency-sampling trajectory, weakening the desirable test-time scaling behavior of ODE sampling. To address this limitation, we introduce AnyFlow, the first any-step video diffusion distillation framework based on flow maps. Instead of distilling a model for only a few fixed sampling steps, AnyFlow optimizes the full ODE sampling trajectory. To this end, we shift the distillation target from endpoint consistency mapping (z_{t}rightarrow z_{0}) to flow-map transition learning (z_{t}rightarrow z_{r}) over arbitrary time intervals. We further propose Flow Map Backward Simulation, which decomposes a full Euler rollout into shortcut flow-map transitions, enabling efficient on-policy distillation that reduces test-time errors (i.e., discretization error in few-step sampling and exposure bias in causal generation). Extensive experiments across both bidirectional and causal architectures, at scales ranging from 1.3B to 14B parameters, demonstrate that AnyFlow achieves performance matches or surpasses consistency-based counterparts in the few-step regime, while scaling with sampling step budgets.

We demonstrate the possibility of learning drone swarm controllers that are zero-shot transferable to real quadrotors via large-scale multi-agent end-to-end reinforcement learning. We train policies parameterized by neural networks that are capable of controlling individual drones in a swarm in a fully decentralized manner. Our policies, trained in simulated environments with realistic quadrotor physics, demonstrate advanced flocking behaviors, perform aggressive maneuvers in tight formations while avoiding collisions with each other, break and re-establish formations to avoid collisions with moving obstacles, and efficiently coordinate in pursuit-evasion tasks. We analyze, in simulation, how different model architectures and parameters of the training regime influence the final performance of neural swarms. We demonstrate the successful deployment of the model learned in simulation to highly resource-constrained physical quadrotors performing station keeping and goal swapping behaviors. Code and video demonstrations are available on the project website at https://sites.google.com/view/swarm-rl.

Open time-series forecasting (TSF) benchmarks cover retail, energy, weather, and traffic, but supply-chain logistics remains underserved. We introduce ISOMORPH, the first public digital twin of a multi-echelon logistics network with fully interpretable, user-configurable parameters and modular topology, demand process, and control rules. The simulator advances a directed routing graph in discrete time: demand arrives at the destination, is served from stock or recorded as backlog, and triggers replenishment through the network. The state vector tracks per-node on-hand inventory with outstanding orders, in-transit shipments, and a smoothed demand estimate, so the dynamics close as a Markov chain on a tractable state space whose transition kernel acts linearly on the empirical distribution of the state. The released data reproduces the bullwhip effect at empirically consistent magnitudes, and three conservation laws encoded in the Markov chain serve as verification tools when users extend the simulator. We release datasets at two catalogue scales (C=50 and C=200) with six scenario sweeps producing 30 additional rollouts and 20 Latin-hypercube perturbations, exhibiting dynamics absent from fixed TSF benchmarks: variance amplification, cascading bottlenecks, regime shifts, and cross-channel coupling through shared macro shocks. Zero-shot evaluation of four foundation models (Chronos, Moirai, TimesFM, Lag-Llama) shows MASE values exceeding public GIFT-Eval references at low-to-moderate horizons, supporting incorporation into existing benchmarks. The same pairing produces forecast confidence bands via Latin-hypercube perturbation of demand-side knobs, forward UQ from parameter uncertainty unavailable on standard TSF datasets, demonstrating that foundation models can serve as fast surrogates for the digital twin's forward UQ. Code (MIT): https://github.com/tuhinsahai/ISOMORPH.

We study compute efficiency of LLM training when using different parameterizations, i.e., rules for adjusting model and optimizer hyperparameters (HPs) as model size changes. Some parameterizations fail to transfer optimal base HPs (such as learning rate) across changes in model depth, requiring practitioners to either re-tune these HPs as they scale up (expensive), or accept sub-optimal training when re-tuning is prohibitive. Even when they achieve HP transfer, we develop theory to show parameterizations may still exist in the lazy learning regime where layers learn only features close to their linearization, preventing effective use of depth and nonlinearity. Finally, we identify and adopt the parameterization we call CompleteP that achieves both depth-wise HP transfer and non-lazy learning in all layers. CompleteP enables a wider range of model width/depth ratios to remain compute-efficient, unlocking shapes better suited for different hardware settings and operational contexts. Moreover, CompleteP enables 12-34% compute efficiency improvements over the prior state-of-the-art.

A natural strategy for continual learning is to weigh a Bayesian ensemble of fixed functions. This suggests that if a (single) neural network could be interpreted as an ensemble, one could design effective algorithms that learn without forgetting. To realize this possibility, we observe that a neural network classifier with N parameters can be interpreted as a weighted ensemble of N classifiers, and that in the lazy regime limit these classifiers are fixed throughout learning. We call these classifiers the neural tangent experts and show they output valid probability distributions over the labels. We then derive the likelihood and posterior probability of each expert given past data. Surprisingly, the posterior updates for these experts are equivalent to a scaled and projected form of stochastic gradient descent (SGD) over the network weights. Away from the lazy regime, networks can be seen as ensembles of adaptive experts which improve over time. These results offer a new interpretation of neural networks as Bayesian ensembles of experts, providing a principled framework for understanding and mitigating catastrophic forgetting in continual learning settings.

Recent studies of gradient descent with large step sizes have shown that there is often a regime with an initial increase in the largest eigenvalue of the loss Hessian (progressive sharpening), followed by a stabilization of the eigenvalue near the maximum value which allows convergence (edge of stability). These phenomena are intrinsically non-linear and do not happen for models in the constant Neural Tangent Kernel (NTK) regime, for which the predictive function is approximately linear in the parameters. As such, we consider the next simplest class of predictive models, namely those that are quadratic in the parameters, which we call second-order regression models. For quadratic objectives in two dimensions, we prove that this second-order regression model exhibits progressive sharpening of the NTK eigenvalue towards a value that differs slightly from the edge of stability, which we explicitly compute. In higher dimensions, the model generically shows similar behavior, even without the specific structure of a neural network, suggesting that progressive sharpening and edge-of-stability behavior aren't unique features of neural networks, and could be a more general property of discrete learning algorithms in high-dimensional non-linear models.

A biological neural network in the cortex forms a neural field. Neurons in the field have their own receptive fields, and connection weights between two neurons are random but highly correlated when they are in close proximity in receptive fields. In this paper, we investigate such neural fields in a multilayer architecture to investigate the supervised learning of the fields. We empirically compare the performances of our field model with those of randomly connected deep networks. The behavior of a randomly connected network is investigated on the basis of the key idea of the neural tangent kernel regime, a recent development in the machine learning theory of over-parameterized networks; for most randomly connected neural networks, it is shown that global minima always exist in their small neighborhoods. We numerically show that this claim also holds for our neural fields. In more detail, our model has two structures: i) each neuron in a field has a continuously distributed receptive field, and ii) the initial connection weights are random but not independent, having correlations when the positions of neurons are close in each layer. We show that such a multilayer neural field is more robust than conventional models when input patterns are deformed by noise disturbances. Moreover, its generalization ability can be slightly superior to that of conventional models.

Current state-of-the-art solutions for motion capture from a single camera are optimization driven: they optimize the parameters of a 3D human model so that its re-projection matches measurements in the video (e.g. person segmentation, optical flow, keypoint detections etc.). Optimization models are susceptible to local minima. This has been the bottleneck that forced using clean green-screen like backgrounds at capture time, manual initialization, or switching to multiple cameras as input resource. In this work, we propose a learning based motion capture model for single camera input. Instead of optimizing mesh and skeleton parameters directly, our model optimizes neural network weights that predict 3D shape and skeleton configurations given a monocular RGB video. Our model is trained using a combination of strong supervision from synthetic data, and self-supervision from differentiable rendering of (a) skeletal keypoints, (b) dense 3D mesh motion, and (c) human-background segmentation, in an end-to-end framework. Empirically we show our model combines the best of both worlds of supervised learning and test-time optimization: supervised learning initializes the model parameters in the right regime, ensuring good pose and surface initialization at test time, without manual effort. Self-supervision by back-propagating through differentiable rendering allows (unsupervised) adaptation of the model to the test data, and offers much tighter fit than a pretrained fixed model. We show that the proposed model improves with experience and converges to low-error solutions where previous optimization methods fail.

How can prior knowledge on the transformation invariances of a domain be incorporated into the architecture of a neural network? We propose Equivariant Transformers (ETs), a family of differentiable image-to-image mappings that improve the robustness of models towards pre-defined continuous transformation groups. Through the use of specially-derived canonical coordinate systems, ETs incorporate functions that are equivariant by construction with respect to these transformations. We show empirically that ETs can be flexibly composed to improve model robustness towards more complicated transformation groups in several parameters. On a real-world image classification task, ETs improve the sample efficiency of ResNet classifiers, achieving relative improvements in error rate of up to 15% in the limited data regime while increasing model parameter count by less than 1%.

Offline optimization is an important task in numerous material engineering domains where online experimentation to collect data is too expensive and needs to be replaced by an in silico maximization of a surrogate of the black-box function. Although such a surrogate can be learned from offline data, its prediction might not be reliable outside the offline data regime, which happens when the surrogate has narrow prediction margin and is (therefore) sensitive to small perturbations of its parameterization. This raises the following questions: (1) how to regulate the sensitivity of a surrogate model; and (2) whether conditioning an offline optimizer with such less sensitive surrogate will lead to better optimization performance. To address these questions, we develop an optimizable sensitivity measurement for the surrogate model, which then inspires a sensitivity-informed regularizer that is applicable to a wide range of offline optimizers. This development is both orthogonal and synergistic to prior research on offline optimization, which is demonstrated in our extensive experiment benchmark.

We consider dense, associative neural-networks trained by a teacher (i.e., with supervision) and we investigate their computational capabilities analytically, via statistical-mechanics of spin glasses, and numerically, via Monte Carlo simulations. In particular, we obtain a phase diagram summarizing their performance as a function of the control parameters such as quality and quantity of the training dataset, network storage and noise, that is valid in the limit of large network size and structureless datasets: these networks may work in a ultra-storage regime (where they can handle a huge amount of patterns, if compared with shallow neural networks) or in a ultra-detection regime (where they can perform pattern recognition at prohibitive signal-to-noise ratios, if compared with shallow neural networks). Guided by the random theory as a reference framework, we also test numerically learning, storing and retrieval capabilities shown by these networks on structured datasets as MNist and Fashion MNist. As technical remarks, from the analytic side, we implement large deviations and stability analysis within Guerra's interpolation to tackle the not-Gaussian distributions involved in the post-synaptic potentials while, from the computational counterpart, we insert Plefka approximation in the Monte Carlo scheme, to speed up the evaluation of the synaptic tensors, overall obtaining a novel and broad approach to investigate supervised learning in neural networks, beyond the shallow limit, in general.

We release Llamion, a family of 14B-parameter open-weight language models obtained by transforming Orion-14B into the standardized Llama-family architecture. The transformation is performed by Efficient Knowledge Preservation for Transformation (KEPT), a recipe that combines (i) Normal Parameter Mapping (NPM) for unchanged modules, (ii) Optimized Parameter Mapping (OPM), a training-free LayerNorm-to-RMSNorm initialization we prove optimal under the near-zero-mean activation regime induced by weight decay, and (iii) Cross-architecture Knowledge Distillation (XKD), an equal-size frozen-teacher distillation that aligns the converted model's outputs with the source model's on any reasonable input distribution. Llamion recovers Orion's behaviour on H6, MT-Bench, and KoMMLU with only ~123M tokens on a single A100 in four days; Llamion-Base reaches 66.87% on KoMMLU, exceeding the next-best entry of the Open Ko LLM Leaderboard by >7.0 absolute points at submission time. Capabilities entirely absent from the transfer corpus (Python programming and 200K-token context handling) survive the architectural transition intact. We release three checkpoints (Base, Chat, LongChat) that load with trust_remote_code=False in the Hugging Face Transformers library.

We introduce rd-spiral, an open-source Python library for simulating 2D reaction-diffusion systems using pseudo-spectral methods. The framework combines FFT-based spatial discretization with adaptive Dormand-Prince time integration, achieving exponential convergence while maintaining pedagogical clarity. We analyze three dynamical regimes: stable spirals, spatiotemporal chaos, and pattern decay, revealing extreme non-Gaussian statistics (kurtosis >96) in stable states. Information-theoretic metrics show 10.7% reduction in activator-inhibitor coupling during turbulence versus 6.5% in stable regimes. The solver handles stiffness ratios >6:1 with features including automated equilibrium classification and checkpointing. Effect sizes (delta=0.37--0.78) distinguish regimes, with asymmetric field sensitivities to perturbations. By balancing computational rigor with educational transparency, rd-spiral bridges theoretical and practical nonlinear dynamics.

This paper introduces a continuous-time stochastic dynamical framework for understanding how large language models (LLMs) may self-amplify latent biases or toxicity through their own chain-of-thought reasoning. The model posits an instantaneous "severity" variable x(t) in [0,1] evolving under a stochastic differential equation (SDE) with a drift term μ(x) and diffusion σ(x). Crucially, such a process can be consistently analyzed via the Fokker--Planck approach if each incremental step behaves nearly Markovian in severity space. The analysis investigates critical phenomena, showing that certain parameter regimes create phase transitions from subcritical (self-correcting) to supercritical (runaway severity). The paper derives stationary distributions, first-passage times to harmful thresholds, and scaling laws near critical points. Finally, it highlights implications for agents and extended LLM reasoning models: in principle, these equations might serve as a basis for formal verification of whether a model remains stable or propagates bias over repeated inferences.

Scaling laws predict loss from compute but not how capabilities interact. We measure the coupling between reasoning and truthfulness across 63 base models from 16 families and find a regime change invisible to loss curves: below a family-dependent critical scale N_c, capabilities anticorrelate; above it, they cooperate. N_c approx 3.5B parameters [2.9B, 13.4B] (bootstrap 95% CI), but model size is not the only variable that determines phase. Architecture, data curation, and training recipe each shift N_c independently: curated training eliminated the coupling dip between Qwen generations (0.025 to 0.830 at matched scale), Gemma-4 at 4B achieves coupling 0.871, characteristic of 13B+ standard-trained models, through distillation and architectural innovation, and Phi at 1B matches web-trained coupling at 10B through data curation alone. Width normalization eliminates the anticorrelation across all tested families, supporting an output-projection bottleneck. Internally, 38 of 40 models show zero competing attention heads. A sparse-regression ODE cross-predicts held-out Llama-2 at 5.6% error. The diagnostic requires no model internals -- only public benchmark scores across a model family. The cooperative regime extends to the frontier (r = +0.72, 34 models, 10 labs). Code, data, and an open-source activation-steering tool for any open-weight model are released alongside an interactive dashboard that diagnoses any model's coupling phase, suggests concrete interventions (data curation, width, benchmark rotation), and provides ODE scaling predictions, frontier diagnostics, and eigenstructure analysis: https://zehenlabs.com/cape/.

Chemical reaction networks can undergo nonequilibrium phase transitions upon variation of external control parameters like the chemical potential of a species. We investigate the flux in the associated chemostats that is proportional to the entropy production and its critical fluctuations within the Schlögl model. Numerical simulations show that the corresponding diffusion coefficient diverges at the critical point as a function of system size. In the vicinity of the critical point, the diffusion coefficient follows a scaling form. We develop an analytical approach based on the chemical Langevin equation and van Kampen's system size expansion that yields the corresponding exponents in the monostable regime. In the bistable regime, we rely on a two-state approximation in order to analytically describe the critical behavior.

We introduce a general, flexible, parametric survival modelling framework which encompasses key shapes of hazard function (constant, increasing, decreasing, up-then-down, down-then-up), various common survival distributions (log-logistic, Burr type XII, Weibull, Gompertz), and includes defective distributions (i.e., cure models). This generality is achieved using four basic distributional parameters: two scale-type parameters and two shape parameters. Generalising to covariate dependence, the scale-type regression components correspond to accelerated failure time (AFT) and proportional hazards (PH) models. Therefore, this general formulation unifies the most popular survival models which allows us to consider the practical value of possible modelling choices for survival data. Furthermore, in line with our proposed flexible baseline distribution, we advocate the use of multi-parameter regression in which more than one distributional parameter depends on covariates - rather than the usual convention of having a single covariate-dependent (scale) parameter. While many choices are available, we suggest introducing covariates through just one or other of the two scale parameters, which covers AFT and PH models, in combination with a `power' shape parameter, which allows for more complex non-AFT/non-PH effects, while the other shape parameter remains covariate-independent, and handles automatic selection of the baseline distribution. We explore inferential issues in simulations, both with and without a covariate, with particular focus on evidence concerning the need, or otherwise, to include both AFT and PH parameters. We illustrate the efficacy of our modelling framework by investigating differences between treatment groups using data from a lung cancer study and a melanoma study. Censoring is accommodated throughout.

Sparse coding, which refers to modeling a signal as sparse linear combinations of the elements of a learned dictionary, has proven to be a successful (and interpretable) approach in applications such as signal processing, computer vision, and medical imaging. While this success has spurred much work on provable guarantees for dictionary recovery when the learned dictionary is the same size as the ground-truth dictionary, work on the setting where the learned dictionary is larger (or over-realized) with respect to the ground truth is comparatively nascent. Existing theoretical results in this setting have been constrained to the case of noise-less data. We show in this work that, in the presence of noise, minimizing the standard dictionary learning objective can fail to recover the elements of the ground-truth dictionary in the over-realized regime, regardless of the magnitude of the signal in the data-generating process. Furthermore, drawing from the growing body of work on self-supervised learning, we propose a novel masking objective for which recovering the ground-truth dictionary is in fact optimal as the signal increases for a large class of data-generating processes. We corroborate our theoretical results with experiments across several parameter regimes showing that our proposed objective also enjoys better empirical performance than the standard reconstruction objective.

In this paper, we examine the thermodynamic behavior of a quantum harmonic oscillator with a position-dependent mass (PDM), where spatial inhomogeneity is modeled through a deformation parameter α. Based on the exact energy spectrum, we explore the resulting thermodynamic quantities and superstatistics. Our findings reveal that increasing α leads to a decrease in entropy and specific heat, reflecting a confinement-induced reduction in the number of accessible states. The partition function and free energy exhibit smooth behavior across all parameter regimes, indicating the absence of critical phase transitions. This study underscores the influence of mass deformation on quantum thermal responses and demonstrates that, while the overall thermodynamic trends are consistent with those reported in the literature, certain distinctive features emerge due to the specific form of the deformation.

We introduce Sven (Singular Value dEsceNt), a new optimization algorithm for neural networks that exploits the natural decomposition of loss functions into a sum over individual data points, rather than reducing the full loss to a single scalar before computing a parameter update. Sven treats each data point's residual as a separate condition to be satisfied simultaneously, using the Moore-Penrose pseudoinverse of the loss Jacobian to find the minimum-norm parameter update that best satisfies all conditions at once. In practice, this pseudoinverse is approximated via a truncated singular value decomposition, retaining only the k most significant directions and incurring a computational overhead of only a factor of k relative to stochastic gradient descent. This is in comparison to traditional natural gradient methods, which scale as the square of the number of parameters. We show that Sven can be understood as a natural gradient method generalized to the over-parametrized regime, recovering natural gradient descent in the under-parametrized limit. On regression tasks, Sven significantly outperforms standard first-order methods including Adam, converging faster and to a lower final loss, while remaining competitive with LBFGS at a fraction of the wall-time cost. We discuss the primary challenge to scaling, namely memory overhead, and propose mitigation strategies. Beyond standard machine learning benchmarks, we anticipate that Sven will find natural application in scientific computing settings where custom loss functions decompose into several conditions.

This paper investigates the boundedness of bilinear pseudo-differential operators with symbols in the Hörmander class BS_{varrho,δ}^m(R^n) in the previously unexplored regime 0 leq varrho < δ< 1. We establish boundedness from H^p(R^n) times H^q(R^n) to L^r(R^n) (with L^r replaced by BMO when p=q=r=infty) under the probably optimal condition on the order $m leq m_varrho(p,q) - nmax{δ-varrho,0}{max{r,2}}, where m_\varrho(p,q) is the critical order in the case 0\leqδ\leq\varrho<1. Furthermore, we develop refined pointwise estimates via sharp maximal functions, establishing that for m \leq -n(1-\varrho)(1{\min\{r_1,2\}}+ 1{\min\{r_2,2\}}) with 1<r_{1},r_{2}<\infty, the bilinear operators satisfy M^sharp T_a(f_1,f_2)(x) lesssim M_{r}(f_1,f_2)(x). This extends the parameter range from the restrictive condition 0 \leq δ\leq \varrho < 1 to the general setting 0 \leq \varrho \leq 1, 0 \leq δ< 1 with δ> \varrho permitted, and generalizes previous results of Park and Tomita to distinct exponent pairs. Consequently, we obtain weighted norm inequalities for bilinear pseudo-differential operators under multilinear A_{p,(r,\infty)}$ weights.

It has been proposed that the superconducting transition temperature T_{c} of an unconventional superconductor with a large pairing scale but strong phase fluctuations can be enhanced by coupling it to a metal. However, the general efficacy of this approach across different parameter regimes remains an open question. Using the dynamical cluster approximation, we study this question in a system composed of an attractive Hubbard layer in the intermediate coupling regime, where the magnitude of the attractive Coulomb interaction |U| is slightly larger than the bandwidth W, hybridized with a noninteracting metallic layer. We find that while the superconducting transition becomes more mean-field-like with increasing interlayer hopping, the superconducting transition temperature T_{c} exhibits a nonmonotonic dependence on the strength of the hybridization t_{perp}. This behavior arises from a reduction of the effective pairing interaction in the correlated layer that out-competes the growth in the intrinsic pair-field susceptibility induced by the coupling to the metallic layer. We find that the largest T_{c} inferred here for the composite system is below the maximum value currently estimated for the isolated negative-U Hubbard model.

Federated learning (FL) is a trending training paradigm to utilize decentralized training data. FL allows clients to update model parameters locally for several epochs, then share them to a global model for aggregation. This training paradigm with multi-local step updating before aggregation exposes unique vulnerabilities to adversarial attacks. Adversarial training is a popular and effective method to improve the robustness of networks against adversaries. In this work, we formulate a general form of federated adversarial learning (FAL) that is adapted from adversarial learning in the centralized setting. On the client side of FL training, FAL has an inner loop to generate adversarial samples for adversarial training and an outer loop to update local model parameters. On the server side, FAL aggregates local model updates and broadcast the aggregated model. We design a global robust training loss and formulate FAL training as a min-max optimization problem. Unlike the convergence analysis in classical centralized training that relies on the gradient direction, it is significantly harder to analyze the convergence in FAL for three reasons: 1) the complexity of min-max optimization, 2) model not updating in the gradient direction due to the multi-local updates on the client-side before aggregation and 3) inter-client heterogeneity. We address these challenges by using appropriate gradient approximation and coupling techniques and present the convergence analysis in the over-parameterized regime. Our main result theoretically shows that the minimum loss under our algorithm can converge to epsilon small with chosen learning rate and communication rounds. It is noteworthy that our analysis is feasible for non-IID clients.

This paper considers the problem of variable selection allowing for parameter instability. It distinguishes between signal and pseudo-signal variables that are correlated with the target variable, and noise variables that are not, and investigate the asymptotic properties of the One Covariate at a Time Multiple Testing (OCMT) method proposed by Chudik et al. (2018) under parameter insatiability. It is established that OCMT continues to asymptotically select an approximating model that includes all the signals and none of the noise variables. Properties of post selection regressions are also investigated, and in-sample fit of the selected regression is shown to have the oracle property. The theoretical results support the use of unweighted observations at the selection stage of OCMT, whilst applying down-weighting of observations only at the forecasting stage. Monte Carlo and empirical applications show that OCMT without down-weighting at the selection stage yields smaller mean squared forecast errors compared to Lasso, Adaptive Lasso, and boosting.

The realization of scalable fault-tolerant quantum computing is expected to hinge on quantum error-correcting codes. In the quest for more efficient quantum fault tolerance, a critical code parameter is the weight of measurements that extract information about errors to enable error correction: as higher measurement weights require higher implementation costs and introduce more errors, it is important in code design to optimize measurement weight. This underlies the surging interest in quantum low-density parity-check (qLDPC) codes, the study of which has primarily focused on the asymptotic (large-code-limit) properties. In this work, we introduce a versatile and computationally efficient approach to stabilizer code weight reduction based on reinforcement learning (RL), which produces new low-weight codes that substantially outperform the state of the art in practically relevant parameter regimes, extending significantly beyond previously accessible small distances. For example, our approach demonstrates savings in physical qubit overhead compared to existing results by 1 to 2 orders of magnitude for weight 6 codes and brings the overhead into a feasible range for near-future experiments. We also investigate the interplay between code parameters using our RL framework, offering new insights into the potential efficiency and power of practically viable coding strategies. Overall, our results demonstrate how RL can effectively advance the crucial yet challenging problem of quantum code discovery and thereby facilitate a faster path to the practical implementation of fault-tolerant quantum technologies.

Which transformer scaling regimes are able to perfectly solve different classes of algorithmic problems? While tremendous empirical advances have been attained by transformer-based neural networks, a theoretical understanding of their algorithmic reasoning capabilities in realistic parameter regimes is lacking. We investigate this question in terms of the network's depth, width, and number of extra tokens for algorithm execution. Our novel representational hierarchy separates 9 algorithmic reasoning problems into classes solvable by transformers in different realistic parameter scaling regimes. We prove that logarithmic depth is necessary and sufficient for tasks like graph connectivity, while single-layer transformers with small embedding dimensions can solve contextual retrieval tasks. We also support our theoretical analysis with ample empirical evidence using the GraphQA benchmark. These results show that transformers excel at many graph reasoning tasks, even outperforming specialized graph neural networks.

We introduce JoyAI-LLM Flash, an efficient Mixture-of-Experts (MoE) language model designed to redefine the trade-off between strong performance and token efficiency in the sub-50B parameter regime. JoyAI-LLM Flash is pretrained on a massive corpus of 20 trillion tokens and further optimized through a rigorous post-training pipeline, including supervised fine-tuning (SFT), Direct Preference Optimization (DPO), and large-scale reinforcement learning (RL) across diverse environments. To improve token efficiency, JoyAI-LLM Flash strategically balances thinking and non-thinking cognitive modes and introduces FiberPO, a novel RL algorithm inspired by fibration theory that decomposes trust-region maintenance into global and local components, providing unified multi-scale stability control for LLM policy optimization. To enhance architectural sparsity, the model comprises 48B total parameters while activating only 2.7B parameters per forward pass, achieving a substantially higher sparsity ratio than contemporary industry leading models of comparable scale. To further improve inference throughput, we adopt a joint training-inference co-design that incorporates dense Multi-Token Prediction (MTP) and Quantization-Aware Training (QAT). We release the checkpoints for both JoyAI-LLM-48B-A3B Base and its post-trained variants on Hugging Face to support the open-source community.

Scaling laws play a central role in the success of Large Language Models (LLMs), enabling the prediction of model performance relative to compute budgets prior to training. While Transformers have been the dominant architecture, recent alternatives such as xLSTM offer linear complexity with respect to context length while remaining competitive in the billion-parameter regime. We conduct a comparative investigation on the scaling behavior of Transformers and xLSTM along the following lines, providing insights to guide future model design and deployment. First, we study the scaling behavior for xLSTM in compute-optimal and over-training regimes using both IsoFLOP and parametric fit approaches on a wide range of model sizes (80M-7B) and number of training tokens (2B-2T). Second, we examine the dependence of optimal model sizes on context length, a pivotal aspect that was largely ignored in previous work. Finally, we analyze inference-time scaling characteristics. Our findings reveal that in typical LLM training and inference scenarios, xLSTM scales favorably compared to Transformers. Notably, xLSTM models consistently Pareto-dominate Transformer models, delivering lower cross-entropy loss for the same compute budget.

We introduce BCAT, a PDE foundation model designed for autoregressive prediction of solutions to two dimensional fluid dynamics problems. Our approach uses a block causal transformer architecture to model next frame predictions, leveraging previous frames as contextual priors rather than relying solely on sub-frames or pixel-based inputs commonly used in image generation methods. This block causal framework more effectively captures the spatial dependencies inherent in nonlinear spatiotemporal dynamics and physical phenomena. In an ablation study, next frame prediction demonstrated a 3.5x accuracy improvement over next token prediction. BCAT is trained on a diverse range of fluid dynamics datasets, including incompressible and compressible Navier-Stokes equations across various geometries and parameter regimes, as well as the shallow-water equations. The model's performance was evaluated on 6 distinct downstream prediction tasks and tested on about 8K trajectories to measure robustness on a variety of fluid dynamics simulations. BCAT achieved an average relative error of 1.18% across all evaluation tasks, outperforming prior approaches on standard benchmarks. With fine-tuning on a turbulence dataset, we show that the method adapts to new settings with more than 40% better accuracy over prior methods.

The transition of an impulsively excited kink oscillation of a solar coronal loop to an oscillation with a stationary amplitude, i.e., the damping pattern, is determined using the low-dimensional self-oscillation model. In the model, the decayless kink oscillations are sustained by the interaction of the oscillating loop with an external quasi-steady flow. The analytical solution is based on the assumption that the combined effect of the effective dissipation, for example, by resonant absorption, and interaction with an external flow, is weak. The effect is characterised by a dimensionless coupling parameter. The damping pattern is found to depend upon the initial amplitude and the coupling parameter. The approximate expression shows a good agreement with a numerical solution of the self-oscillation equation. The plausibility of the established damping pattern is demonstrated by an observational example. Notably, the damping pattern is not exponential, and the characteristic decay time is different from the time determined by the traditionally used exponential damping fit. Implications of this finding for seismology of the solar coronal plasmas are discussed. In particular, it is suggested that a very rapid, in less than the oscillation period, decay of the oscillation to the stationary level, achieved for larger values of the coupling parameter, can explain the relative rareness of the kink oscillation events.

Purpose. Patients with advanced cancer may undergo multiple lines of treatment, switching therapies as their disease progresses. Motivated by a study of metastatic prostate cancer, we develop a microsimulation framework to study therapy sequence. Methods. We propose a discrete-time state transition model to study two lines of anti-cancer therapy. Based on digitized published progression-free survival (PFS) and overall survival (OS) curves, we infer event types (progression or death), and estimate transition probabilities using cumulative incidence functions with competing risks. Our model incorporates within-patient dependence over time, such that response to first-line therapy informs subsequent event probabilities. Parameters governing the degree of within-patient dependence can be used to calibrate the model-based results to those of a target trial. We demonstrate these methods in a study of two therapy sequences for metastatic prostate cancer, where Docetaxel (DCT) and Abiraterone Acetate (AA) are both appropriate for use in either first or second line treatment. We assess costs, Quality-Adjusted Life Years (QALYs) and Incremental Cost Effectiveness Ratio (ICER) for two treatment strategies: DCT then AA vs AA then DCT. Results. Using digitized survival curves from relevant clinical trials, we identified 8.6-13.9% of PFS times that should be categorized as deaths, allowing for estimation of cumulative incidence functions. Models assuming within-patient independence overestimated OS time, corrected with our calibration approach. Correction resulted in meaningful changes in the difference in QALYs between treatment strategies (0.07 vs 0.15) and the ICER (-\76,836/QALY vs -21,030/QALY). Conclusions. Microsimulation models can be successfully used to study cost-effectiveness of therapy sequences, taking care to account correctly for within-patient dependence.

With climate change-related extreme events on the rise, high dimensional Earth observation data presents a unique opportunity for forecasting and understanding impacts on ecosystems. This is, however, impeded by the complexity of processing, visualizing, modeling, and explaining this data. To showcase how this challenge can be met, here we train a convolutional long short-term memory-based architecture on the novel DeepExtremeCubes dataset. DeepExtremeCubes includes around 40,000 long-term Sentinel-2 minicubes (January 2016-October 2022) worldwide, along with labeled extreme events, meteorological data, vegetation land cover, and topography map, sampled from locations affected by extreme climate events and surrounding areas. When predicting future reflectances and vegetation impacts through kernel normalized difference vegetation index, the model achieved an R^2 score of 0.9055 in the test set. Explainable artificial intelligence was used to analyze the model's predictions during the October 2020 Central South America compound heatwave and drought event. We chose the same area exactly one year before the event as counterfactual, finding that the average temperature and surface pressure are generally the best predictors under normal conditions. In contrast, minimum anomalies of evaporation and surface latent heat flux take the lead during the event. A change of regime is also observed in the attributions before the event, which might help assess how long the event was brewing before happening. The code to replicate all experiments and figures in this paper is publicly available at https://github.com/DeepExtremes/txyXAI

We perform a Monte Carlo analysis of the Ising model on many three-dimensional lattices. By means of finite-size scaling we obtain the critical points and determine the scaling dimensions. As expected, the critical exponents agree with the three-dimensional Ising universality class for all models. The irrelevant field, as revealed by the correction-to-scaling amplitudes, appears to be relatively large. Combining the Monte Carlo results for the hydrogen peroxide lattice with those for five other three-dimensional lattices, we obtain a set of data covering a wide range of the irrelevant temperature field. This is helpful in the determination of the parameters describing the corrections to scaling. As a consequence, new results are obtained for the universal parameters describing Ising criticality in three dimensions, with reduced error margins in comparison with earlier Monte Carlo analyses. The critical exponents describing the thermodynamic singularities are determined by the temperature renormalization exponent y_t = 1.58693 (9) and the magnetic renormalization exponent y_h = 2.48178 (5). The corrections to scaling are governed by the irrelevant exponent y_1 = -0.821 (5).

Critical temperature, T_c, and the transition width, ΔT_c, are two primary parameters of the superconducting transition. The latter parameter reflects the superconducting state disturbance originating from the thermodynamic fluctuations, atomic disorder, applied magnetic field, the presence of secondary crystalline phases, applied pressure, etc. Recently, Hirsch and Marsiglio (2020 arXiv:2012.12796) performed an analysis of the transition width in several near-room-temperature superconductors (NRTS) and reported that the reduced transition width, ΔT_c/T_c, in these materials does not follow a conventional trend of transition width broadening on applied magnetic field observed in low- and high-T_c superconductors. Here we present thorough mathematical analysis of the magnetoresistive data, R(T,B), for the high-entropy alloy (ScZrNb)_{0.65}[RhPd]_{0.35} and hydrogen-rich superconductors of Im-3m-H_{3}S, C2/m-LaH_{10} and P63/mmc-CeH_9. We found that the reduced transition width, ΔT_c/T_c, in these materials does follow a conventional broadening trend on applied magnetic field.

We study in detail a one-dimensional lattice model of a continuum, conserved field (mass) that is transferred deterministically between neighbouring random sites. The model falls in a wider class of lattice models capturing the joint effect of random advection and diffusion and encompassing as specific cases, some models studied in the literature, like the Kang-Redner, Kipnis-Marchioro-Presutti, Takayasu-Taguchi, etc. The motivation for our setup comes from a straightforward interpretation as advection of particles in one-dimensional turbulence, but it is also related to a problem of synchronization of dynamical systems driven by common noise. For finite lattices, we study both the coalescence of an initially spread field (interpreted as roughening), and the statistical steady-state properties. We distinguish two main size-dependent regimes, depending on the strength of the diffusion term and on the lattice size. Using numerical simulations and mean-field approach, we study the statistics of the field. For weak diffusion, we unveil a characteristic hierarchical structure of the field. We also connect the model and the iterated function systems concept.

The objective of this study is to evaluate the potential for History Matching (HM) to tune a climate system with multi-scale dynamics. By considering a toy climate model, namely, the two-scale Lorenz96 model and producing experiments in perfect-model setting, we explore in detail how several built-in choices need to be carefully tested. We also demonstrate the importance of introducing physical expertise in the range of parameters, a priori to running HM. Finally we revisit a classical procedure in climate model tuning, that consists of tuning the slow and fast components separately. By doing so in the Lorenz96 model, we illustrate the non-uniqueness of plausible parameters and highlight the specificity of metrics emerging from the coupling. This paper contributes also to bridging the communities of uncertainty quantification, machine learning and climate modeling, by making connections between the terms used by each community for the same concept and presenting promising collaboration avenues that would benefit climate modeling research.

We consider a biological population whose environment varies periodically in time, exhibiting two very different "seasons" : one is favorable and the other one is unfavorable. For monotone differential models with concave nonlinearities, we address the following question: the system's period being fixed, under what conditions does there exist a critical duration for the unfavorable season? By "critical duration" we mean that above some threshold, the population cannot sustain and extincts, while below this threshold, the system converges to a unique periodic and positive solution. We term this a "sharp seasonal threshold property" (SSTP, for short). Building upon a previous result, we obtain sufficient conditions for SSTP in any dimension and apply our criterion to a two-dimensional model featuring juvenile and adult populations of insects.

This paper studies Kernel density estimation for a high-dimensional distribution rho(x). Traditional approaches have focused on the limit of large number of data points n and fixed dimension d. We analyze instead the regime where both the number n of data points y_i and their dimensionality d grow with a fixed ratio alpha=(log n)/d. Our study reveals three distinct statistical regimes for the kernel-based estimate of the density hat rho_h^{D}(x)=1{n h^d}sum_{i=1}^n Kleft(x-y_i{h}right), depending on the bandwidth h: a classical regime for large bandwidth where the Central Limit Theorem (CLT) holds, which is akin to the one found in traditional approaches. Below a certain value of the bandwidth, h_{CLT}(alpha), we find that the CLT breaks down. The statistics of hat rho_h^{D}(x) for a fixed x drawn from rho(x) is given by a heavy-tailed distribution (an alpha-stable distribution). In particular below a value h_G(alpha), we find that hat rho_h^{D}(x) is governed by extreme value statistics: only a few points in the database matter and give the dominant contribution to the density estimator. We provide a detailed analysis for high-dimensional multivariate Gaussian data. We show that the optimal bandwidth threshold based on Kullback-Leibler divergence lies in the new statistical regime identified in this paper. Our findings reveal limitations of classical approaches, show the relevance of these new statistical regimes, and offer new insights for Kernel density estimation in high-dimensional settings.

Common-pool resource governance requires users to cooperate and avoid overexploitation, but defection and free-riding often undermine cooperation. We model a human-environmental system that integrates dynamics of resource and users' strategies. The resource follows a logistic function that depends on natural growth rate, carrying capacity, and extraction rates of cooperators and defectors. The users' strategies evolve according to different processes that capture effects of payoff, resource, and noise. We analyze the feedback between resource availability and strategic adaptation, and explores the conditions for the emergence and maintenance of cooperation. We find different processes lead to different regimes of equilibrium solutions and resource levels depending on the parameter configuration and initial conditions. We also show that some processes can enhance the sustainability of the resource by making the users more responsive to the resource scarcity. The paper advances the understanding of human-environmental system and offers insights for resource governance policies and interventions.

Catastrophic regime shifts in complex natural systems may be averted through advanced detection. Recent work has provided a proof-of-principle that many systems approaching a catastrophic transition may be identified through the lens of early warning indicators such as rising variance or increased return times. Despite widespread appreciation of the difficulties and uncertainty involved in such forecasts, proposed methods hardly ever characterize their expected error rates. Without the benefits of replicates, controls, or hindsight, applications of these approaches must quantify how reliable different indicators are in avoiding false alarms, and how sensitive they are to missing subtle warning signs. We propose a model based approach in order to quantify this trade-off between reliability and sensitivity and allow comparisons between different indicators. We show these error rates can be quite severe for common indicators even under favorable assumptions, and also illustrate how a model-based indicator can improve this performance. We demonstrate how the performance of an early warning indicator varies in different data sets, and suggest that uncertainty quantification become a more central part of early warning predictions.

We develop a rigorous walk-forward validation framework for algorithmic trading designed to mitigate overfitting and lookahead bias. Our methodology combines interpretable hypothesis-driven signal generation with reinforcement learning and strict out-of-sample testing. The framework enforces strict information set discipline, employs rolling window validation across 34 independent test periods, maintains complete interpretability through natural language hypothesis explanations, and incorporates realistic transaction costs and position constraints. Validating five market microstructure patterns across 100 US equities from 2015 to 2024, the system yields modest annualized returns (0.55%, Sharpe ratio 0.33) with exceptional downside protection (maximum drawdown -2.76%) and market-neutral characteristics (beta = 0.058). Performance exhibits strong regime dependence, generating positive returns during high-volatility periods (0.60% quarterly, 2020-2024) while underperforming in stable markets (-0.16%, 2015-2019). We report statistically insignificant aggregate results (p-value 0.34) to demonstrate a reproducible, honest validation protocol that prioritizes interpretability and extends naturally to advanced hypothesis generators, including large language models. The key empirical finding reveals that daily OHLCV-based microstructure signals require elevated information arrival and trading activity to function effectively. The framework provides complete mathematical specifications and open-source implementation, establishing a template for rigorous trading system evaluation that addresses the reproducibility crisis in quantitative finance research. For researchers, practitioners, and regulators, this work demonstrates that interpretable algorithmic trading strategies can be rigorously validated without sacrificing transparency or regulatory compliance.

Chaos and unpredictability are traditionally synonymous, yet large-scale machine learning methods recently have demonstrated a surprising ability to forecast chaotic systems well beyond typical predictability horizons. However, recent works disagree on whether specialized methods grounded in dynamical systems theory, such as reservoir computers or neural ordinary differential equations, outperform general-purpose large-scale learning methods such as transformers or recurrent neural networks. These prior studies perform comparisons on few individually-chosen chaotic systems, thereby precluding robust quantification of how statistical modeling choices and dynamical invariants of different chaotic systems jointly determine empirical predictability. Here, we perform the largest to-date comparative study of forecasting methods on the classical problem of forecasting chaos: we benchmark 24 state-of-the-art forecasting methods on a crowdsourced database of 135 low-dimensional systems with 17 forecast metrics. We find that large-scale, domain-agnostic forecasting methods consistently produce predictions that remain accurate up to two dozen Lyapunov times, thereby accessing a new long-horizon forecasting regime well beyond classical methods. We find that, in this regime, accuracy decorrelates with classical invariant measures of predictability like the Lyapunov exponent. However, in data-limited settings outside the long-horizon regime, we find that physics-based hybrid methods retain a comparative advantage due to their strong inductive biases.

In this article, we consider a newly proposed parameterization of the viscosity coefficient zeta, specifically zeta=zeta_0 {Omega^s_m} H , where zeta_0 = zeta_0{{Omega^s_{m_0}}} within the coincident f(Q) gravity formalism. We consider a non-linear function f(Q)= -Q +alpha Q^n, where alpha and n are arbitrary model parameters, which is a power-law correction to the STEGR scenario. We find an autonomous system by invoking the dimensionless density parameters as the governing phase-space variables. We discuss the physical significance of the model corresponding to the parameter choices n=-1 and n=2 along with the exponent choices s=0, 0.5, and 1.05. We find that model I shows the stable de-Sitter type or stable phantom type (depending on the choice of exponent s) behavior with no transition epoch, whereas model II shows the evolutionary phase from the radiation epoch to the accelerated de-Sitter epoch via passing through the matter-dominated epoch. Hence, we conclude that model I provides a good description of the late-time cosmology but fails to describe the transition epoch, whereas model II modifies the description in the context of the early universe and provides a good description of the matter and radiation era along with the transition phase.

We investigate a new dynamical system that describes tumor-host interaction. The equation that describes the untreated tumor growth is based on non-extensive statistical mechanics. Recently, this model has been shown to fit successfully exponential, Gompertz, logistic, and power-law tumor growths. We have been able to include as many hallmarks of cancer as possible. We study also the dynamic response of cancer under therapy. Using our model, we can make predictions about the different outcomes when we change the parameters, and/or the initial conditions. We can determine the importance of different factors to influence tumor growth. We discover synergistic therapeutic effects of different treatments and drugs. Cancer is generally untreatable using conventional monotherapy. We consider conventional therapies, oncogene-targeted therapies, tumor-suppressors gene-targeted therapies, immunotherapies, anti-angiogenesis therapies, virotherapy, among others. We need therapies with the potential to target both tumor cells and the tumors' microenvironment. Drugs that target oncogenes and tumor-suppressor genes can be effective in the treatment of some cancers. However, most tumors do reoccur. We have found that the success of the new therapeutic agents can be seen when used in combination with other cancer-cell-killing therapies. Our results have allowed us to design a combinational therapy that can lead to the complete eradication of cancer.

Hyperparameter tuning is one of the essential steps to guarantee the convergence of machine learning models. We argue that intuition about the optimal choice of hyperparameters for stochastic gradient descent can be obtained by studying a neural network's phase diagram, in which each phase is characterised by distinctive dynamics of the singular values of weight matrices. Taking inspiration from disordered systems, we start from the observation that the loss landscape of a multilayer neural network with mean squared error can be interpreted as a disordered system in feature space, where the learnt features are mapped to soft spin degrees of freedom, the initial variance of the weight matrices is interpreted as the strength of the disorder, and temperature is given by the ratio of the learning rate and the batch size. As the model is trained, three phases can be identified, in which the dynamics of weight matrices is qualitatively different. Employing a Langevin equation for stochastic gradient descent, previously derived using Dyson Brownian motion, we demonstrate that the three dynamical regimes can be classified effectively, providing practical guidance for the choice of hyperparameters of the optimiser.

We present a method for parametrizing sub-grid processes in the Shallow Water equations. We define coarse variables and local spatial averages and use a feed-forward neural network to learn sub-grid fluxes. Our method results in a local parametrization that uses a four-point computational stencil, which has several advantages over globally coupled parametrizations. We demonstrate numerically that our method improves energy balance in long-term turbulent simulations and also accurately reproduces individual solutions. The neural network parametrization can be easily combined with flux limiting to reduce oscillations near shocks. More importantly, our method provides reliable parametrizations, even in dynamical regimes that are not included in the training data.

A compartmental deterministic model that allows (1) immunity from two stages of infection and carriage, and (2) disease induced death, is used in studying the dynamics of meningitis epidemic process in a closed population. It allows for difference in the transmission rate of infection to a susceptible by a carrier and an infective. It is generalized to allow a proportion ({\phi}) of those susceptibles infected to progress directly to infectives in stage I. Both models are used in this study. The threshold conditions for the spread of carrier and infectives in stage I are derived for the two models. Sensitivity analysis is performed on the reproductive number derived from the next generation matrix. The case-carrier ratio profile for various parameters and threshold values are shown. So also are the graphs of the total number ever infected as influenced by {\epsilon} and {\phi}. The infection transmission rate (eta), the odds in favor of a carrier, over an infective, in transmitting an infection to a susceptible ({\epsilon}) and the carrier conversion rate ({\phi}) to an infective in stage I, are identified as key parameters that should be subject of attention for any control intervention strategy. The case-carrier ratio profiles provide evidence of a critical case-carrier ratio attained before the number of reported cases grows to an epidemic level. They also provide visual evidence of epidemiological context, in this case, epidemic incidence (in later part of dry season) and endemic incidence (during rainy season). Results from total proportion ever infected suggest that the model, in which {\phi}=0 obtained, can adequately represent, in essence, the generalized model for this study.

Global climate models (GCMs) are the main tools for understanding and predicting climate change. However, due to limited numerical resolutions, these models suffer from major structural uncertainties; e.g., they cannot resolve critical processes such as small-scale eddies in atmospheric and oceanic turbulence. Thus, such small-scale processes have to be represented as a function of the resolved scales via closures (parametrization). The accuracy of these closures is particularly important for capturing climate extremes. Traditionally, such closures are based on heuristics and simplifying assumptions about the unresolved physics. Recently, supervised-learned closures, trained offline on high-fidelity data, have been shown to outperform the classical physics-based closures. However, this approach requires a significant amount of high-fidelity training data and can also lead to instabilities. Reinforcement learning is emerging as a potent alternative for developing such closures as it requires only low-order statistics and leads to stable closures. In Scientific Multi-Agent Reinforcement Learning (SMARL) computational elements serve a dual role of discretization points and learning agents. We leverage SMARL and fundamentals of turbulence physics to learn closures for prototypes of atmospheric and oceanic turbulence. The policy is trained using only the enstrophy spectrum, which is nearly invariant and can be estimated from a few high-fidelity samples (these few samples are far from enough for supervised/offline learning). We show that these closures lead to stable low-resolution simulations that, at a fraction of the cost, can reproduce the high-fidelity simulations' statistics, including the tails of the probability density functions. The results demonstrate the high potential of SMARL for closure modeling for GCMs, especially in the regime of scarce data and indirect observations.

We investigate the time evolution of the Boltzmann entropy of a dilute gas of N particles, N>>1, as it undergoes a free expansion doubling its volume. The microstate of the system, a point in the 4N dimensional phase space, changes in time via Hamiltonian dynamics. Its entropy, at any time t, is given by the logarithm of the phase space volume of all the microstates giving rise to its macrostate at time t. The macrostates that we consider are defined by coarse graining the one-particle phase space into cells Δ_α. The initial and final macrostates of the system are equilibrium states in volumes V and 2V, with the same energy E and particle number N. Their entropy per particle is given, for sufficiently large systems, by the thermodynamic entropy as a function of the particle and energy density, whose leading term is independent of the size of the Δ_α. The intermediate (non-equilibrium) entropy does however depend on the size of the cells Δ_α. Its change with time is due to (i) dispersal in physical space from free motion and to (ii) the collisions between particles which change their velocities. The former depends strongly on the size of the velocity coarse graining Δv: it produces entropy at a rate proportional to Δv. This dependence is investigated numerically and analytically for a dilute two-dimensional gas of hard discs. It becomes significant when the mean free path between collisions is of the same order or larger than the length scale of the initial spatial inhomogeneity. In the opposite limit, the rate of entropy production is essentially independent of Δv and is given by the Boltzmann equation for the limit Δvrightarrow 0. We show that when both processes are active the time dependence of the entropy has a scaling form involving the ratio of the rates of its production by the two processes.

The friction coefficient of a particle can depend on its position as it does when the particle is near a wall. We formulate the dynamics of particles with such state-dependent friction coefficients in terms of a general Langevin equation with multiplicative noise, whose evaluation requires the introduction of specific rules. Two common conventions, the Ito and the Stratonovich, provide alternative rules for evaluation of the noise, but other conventions are possible. We show the requirement that a particle's distribution function approach the Boltzmann distribution at long times dictates that a drift term must be added to the Langevin equation. This drift term is proportional to the derivative of the diffusion coefficient times a factor that depends on the convention used to define the multiplicative noise. We explore the consequences of this result in a number examples with spatially varying diffusion coefficients. We also derive path integral representations for arbitrary interpretation of the noise, and use it in a perturbative study of correlations in a simple system.

The realization that complex systems such as ecological communities can collapse or shift regimes suddenly and without rapid external forcing poses a serious challenge to our understanding and management of the natural world. The potential to identify early warning signals that would allow researchers and managers to predict such events before they happen has therefore been an invaluable discovery that offers a way forward in spite of such seemingly unpredictable behavior. Research into early warning signals has demonstrated that it is possible to define and detect such early warning signals in advance of a transition in certain contexts. Here we describe the pattern emerging as research continues to explore just how far we can generalize these results. A core of examples emerges that shares three properties: the phenomenon of rapid regime shifts, a pattern of 'critical slowing down' that can be used to detect the approaching shift, and a mechanism of bifurcation driving the sudden change. As research has expanded beyond these core examples, it is becoming clear that not all systems that show regime shifts exhibit critical slowing down, or vice versa. Even when systems exhibit critical slowing down, statistical detection is a challenge. We review the literature that explores these edge cases and highlight the need for (a) new early warning behaviors that can be used in cases where rapid shifts do not exhibit critical slowing down, (b) the development of methods to identify which behavior might be an appropriate signal when encountering a novel system; bearing in mind that a positive indication for some systems is a negative indication in others, and (c) statistical methods that can distinguish between signatures of early warning behaviors and noise.

Most of the world's offshore wind resource lies in waters too deep for fixed-bottom foundations, making floating offshore wind turbines (FOWTs) essential for deep-water deployment. As the industry scales toward 22 MW class designs, tower fatigue becomes increasingly critical because larger structures amplify the coupled aero-hydro-servo-elastic loads induced by continuous wind and wave excitation. Accurate fatigue-damage prediction is therefore central to certification, design optimization, and cost reduction. Yet the field lacks a shared surrogate benchmark: studies report different simulations, splits, and metrics, making methods difficult to compare. We present FLOATBench, a public tabular benchmark with 582{,}120 per-section fatigue-damage labels across three 22 MW FOWT tower geometries, derived from 19{,}404 high-fidelity OpenFAST simulations across the three towers (6{,}468 per tower: 1{,}078 aligned wind/wave operating points times six turbulence seeds), labeled at 30 cross-sections per tower. FLOATBench includes a regime-aware alpha-shape partition of the joint wind/wave operating envelope, stratifying test points into in-train, interpolation, and extrapolation regimes. It is paired with a reproducible evaluation harness covering three protocol levels: random validation (E1), within-tower regime-aware evaluation (E2), and cross-tower transfer (E3). The regime-aware protocol reveals rank shifts between global and extrapolation performance that random-split leaderboards cannot detect. To the authors' knowledge, FLOATBench is the first FOWT fatigue benchmark for tabular surrogate modeling, and offers an evaluation protocol that generalizes to engineering surrogates defined over physical operating envelopes. Dataset and code available at: https://github.com/Joao97ribeiro/FLOATBench.

Data pruning algorithms are commonly used to reduce the memory and computational cost of the optimization process. Recent empirical results reveal that random data pruning remains a strong baseline and outperforms most existing data pruning methods in the high compression regime, i.e., where a fraction of 30% or less of the data is kept. This regime has recently attracted a lot of interest as a result of the role of data pruning in improving the so-called neural scaling laws; in [Sorscher et al.], the authors showed the need for high-quality data pruning algorithms in order to beat the sample power law. In this work, we focus on score-based data pruning algorithms and show theoretically and empirically why such algorithms fail in the high compression regime. We demonstrate ``No Free Lunch" theorems for data pruning and present calibration protocols that enhance the performance of existing pruning algorithms in this high compression regime using randomization.

In theoretical neuroscience, recent work leverages deep learning tools to explore how some network attributes critically influence its learning dynamics. Notably, initial weight distributions with small (resp. large) variance may yield a rich (resp. lazy) regime, where significant (resp. minor) changes to network states and representation are observed over the course of learning. However, in biology, neural circuit connectivity could exhibit a low-rank structure and therefore differs markedly from the random initializations generally used for these studies. As such, here we investigate how the structure of the initial weights -- in particular their effective rank -- influences the network learning regime. Through both empirical and theoretical analyses, we discover that high-rank initializations typically yield smaller network changes indicative of lazier learning, a finding we also confirm with experimentally-driven initial connectivity in recurrent neural networks. Conversely, low-rank initialization biases learning towards richer learning. Importantly, however, as an exception to this rule, we find lazier learning can still occur with a low-rank initialization that aligns with task and data statistics. Our research highlights the pivotal role of initial weight structures in shaping learning regimes, with implications for metabolic costs of plasticity and risks of catastrophic forgetting.

A Mamba state-space model trained only for next-step prediction appears to recover Granger-causal structure through a simple readout S = |W_{out} W_{in}|, with early experiments suggesting the phenomenon generalized across architectures and benefited from interventional data at p < 10^{-5}. We package the protocol used to test that claim -- standardized synthetic generators (VAR/Lorenz/CauseMe-style), three intervention semantics (do(X=c), soft-noise, random-forcing), edge-provenance cards on three real datasets, and size-matched control arms -- as a reusable falsification benchmark, and walk the claim through it in five stages. The method-level claim does not survive: (i) a plain linear bottleneck does as well or better; (ii) tuned Lasso beats the bottleneck on synthetic CauseMe-style benchmarks, and on Lorenz-96 (the only real benchmark with unambiguous ground truth) classical PCMCI and Granger lead a tight cluster in which the bottleneck trails; (iii) the headline intervention advantage is roughly 60% a sample-size confound, and the residual disappears under standard do(X=c) interventions, surviving only under a non-standard random-forcing scheme; (iv) even that residual reproduces, with a larger effect, in classical bivariate Granger -- the effect is method-agnostic. What survives is a narrow characterization result; the benchmark is the lasting artifact, and each stage above is one of its control arms.

Approximations based on moment-closure (MA) are commonly used to obtain estimates of the mean molecule numbers and of the variance of fluctuations in the number of molecules of chemical systems. The advantage of this approach is that it can be far less computationally expensive than exact stochastic simulations of the chemical master equation. Here we numerically study the conditions under which the MA equations yield results reflecting the true stochastic dynamics of the system. We show that for bistable and oscillatory chemical systems with deterministic initial conditions, the solution of the MA equations can be interpreted as a valid approximation to the true moments of the CME, only when the steady-state mean molecule numbers obtained from the chemical master equation fall within a certain finite range. The same validity criterion for monostable systems implies that the steady-state mean molecule numbers obtained from the chemical master equation must be above a certain threshold. For mean molecule numbers outside of this range of validity, the MA equations lead to either qualitatively wrong oscillatory dynamics or to unphysical predictions such as negative variances in the molecule numbers or multiple steady-state moments of the stationary distribution as the initial conditions are varied. Our results clarify the range of validity of the MA approach and show that pitfalls in the interpretation of the results can only be overcome through the systematic comparison of the solutions of the MA equations of a certain order with those of higher orders.

Early warning signals have been proposed to forecast the possibility of a critical transition, such as the eutrophication of a lake, the collapse of a coral reef, or the end of a glacial period. Because such transitions often unfold on temporal and spatial scales that can be difficult to approach by experimental manipulation, research has often relied on historical observations as a source of natural experiments. Here we examine a critical difference between selecting systems for study based on the fact that we have observed a critical transition and those systems for which we wish to forecast the approach of a transition. This difference arises by conditionally selecting systems known to experience a transition of some sort and failing to account for the bias this introduces -- a statistical error often known as the Prosecutor's Fallacy. By analysing simulated systems that have experienced transitions purely by chance, we reveal an elevated rate of false positives in common warning signal statistics. We further demonstrate a model-based approach that is less subject to this bias than these more commonly used summary statistics. We note that experimental studies with replicates avoid this pitfall entirely.

We consider the solvable neural scaling model with three parameters: data complexity, target complexity, and model-parameter-count. We use this neural scaling model to derive new predictions about the compute-limited, infinite-data scaling law regime. To train the neural scaling model, we run one-pass stochastic gradient descent on a mean-squared loss. We derive a representation of the loss curves which holds over all iteration counts and improves in accuracy as the model parameter count grows. We then analyze the compute-optimal model-parameter-count, and identify 4 phases (+3 subphases) in the data-complexity/target-complexity phase-plane. The phase boundaries are determined by the relative importance of model capacity, optimizer noise, and embedding of the features. We furthermore derive, with mathematical proof and extensive numerical evidence, the scaling-law exponents in all of these phases, in particular computing the optimal model-parameter-count as a function of floating point operation budget.

We initiate the study of statistical inference and A/B testing for first-price pacing equilibria (FPPE). The FPPE model captures the dynamics resulting from large-scale first-price auction markets where buyers use pacing-based budget management. Such markets arise in the context of internet advertising, where budgets are prevalent. We propose a statistical framework for the FPPE model, in which a limit FPPE with a continuum of items models the long-run steady-state behavior of the auction platform, and an observable FPPE consisting of a finite number of items provides the data to estimate primitives of the limit FPPE, such as revenue, Nash social welfare (a fair metric of efficiency), and other parameters of interest. We develop central limit theorems and asymptotically valid confidence intervals. Furthermore, we establish the asymptotic local minimax optimality of our estimators. We then show that the theory can be used for conducting statistically valid A/B testing on auction platforms. Numerical simulations verify our central limit theorems, and empirical coverage rates for our confidence intervals agree with our theory.

Systemic financial risk refers to the simultaneous failure or destabilization of multiple financial institutions, often triggered by contagion mechanisms or common exposures to shocks. In this paper, we present a dynamical model of bank leverage (the ratio of asset holdings to equity) a quantity that both reflects and drives risk dynamics. We model how banks, constrained by Value-at-Risk (VaR) regulations, adjust their leverage in response to changes in the price of a single asset, assumed to be held in fixed proportion across banks. This leverage-targeting behavior introduces a procyclical feedback loop between asset prices and leverage. In the dynamics, this can manifest as logistic-like behavior with a rich bifurcation structure across model parameters. By analyzing these coupled dynamics in both isolated and interconnected bank models, we outline a framework for understanding how systemic risk can emerge from seemingly rational micro-level behavior.

Using a 100-day symmetric window around the January 2025 U.S. presidential inauguration, non-parametric statistical methods with bootstrap resampling (10,000 iterations) analyze distributional properties and anomalies. Results indicate a statistically significant 3.61\% Indonesian rupiah depreciation post-inauguration, with a large effect size (Cliff's Delta = -0.9224, CI: [-0.9727, -0.8571]). Central tendency shifted markedly, yet volatility remained stable (variance ratio = 0.9061, p = 0.504). Four significant anomalies exhibiting temporal clustering are detected. These findings provide quantitative evidence of political transition effects on emerging market currencies, highlighting implications for monetary policy and currency risk management.

Time series forecasting is critical across finance, healthcare, and cloud computing, yet progress is constrained by a fundamental bottleneck: the scarcity of large-scale, high-quality benchmarks. To address this gap, we introduce QuitoBench, a regime-balanced benchmark for time series forecasting with coverage across eight trendtimesseasonalitytimesforecastability (TSF) regimes, designed to capture forecasting-relevant properties rather than application-defined domain labels. The benchmark is built upon Quito, a billion-scale time series corpus of application traffic from Alipay spanning nine business domains. Benchmarking 10 models from deep learning, foundation models, and statistical baselines across 232,200 evaluation instances, we report four key findings: (i) a context-length crossover where deep learning models lead at short context (L=96) but foundation models dominate at long context (L ge 576); (ii) forecastability is the dominant difficulty driver, producing a 3.64 times MAE gap across regimes; (iii) deep learning models match or surpass foundation models at 59 times fewer parameters; and (iv) scaling the amount of training data provides substantially greater benefit than scaling model size for both model families. These findings are validated by strong cross-benchmark and cross-metric consistency. Our open-source release enables reproducible, regime-aware evaluation for time series forecasting research.

Dynamic treatment regimes (DTRs) are personalized, adaptive, multi-stage treatment plans that adapt treatment decisions both to an individual's initial features and to intermediate outcomes and features at each subsequent stage, which are affected by decisions in prior stages. Examples include personalized first- and second-line treatments of chronic conditions like diabetes, cancer, and depression, which adapt to patient response to first-line treatment, disease progression, and individual characteristics. While existing literature mostly focuses on estimating the optimal DTR from offline data such as from sequentially randomized trials, we study the problem of developing the optimal DTR in an online manner, where the interaction with each individual affect both our cumulative reward and our data collection for future learning. We term this the DTR bandit problem. We propose a novel algorithm that, by carefully balancing exploration and exploitation, is guaranteed to achieve rate-optimal regret when the transition and reward models are linear. We demonstrate our algorithm and its benefits both in synthetic experiments and in a case study of adaptive treatment of major depressive disorder using real-world data.

The main aim of this work is to use a model-independent approach, along with late-time observational probes, to reconstruct the dark energy (DE) equation of state w_{rm DE}(z). Our analysis showed that, for a late time universe, w_{rm DE} deviates from being a constant but in contrast exhibits an oscillatory behavior, hence both quintessence (w_{rm DE}> -1) and phantom (w_{rm DE} < -1) regimes are equally allowed. In order to portray this oscillatory behavior, we explored various parametrizations for the equation of state and identified the closest approximation based on the goodness of fit with the data and the Bayesian evidence analysis. Our findings indicated that while all considered oscillating DE parametrizations provided a better fit to the data, compared to the cosmological constant, they are penalized in the Bayesian evidence analysis due to the additional free parameters. Overall, the present article demonstrates that in the low redshift regime, the equation of state of the DE prefers to be dynamical and oscillating. We anticipate that future cosmological probes will take a stand in this direction.

We address the problem of optimizing a Brownian motion. We consider a (random) realization W of a Brownian motion with input space in [0,1]. Given W, our goal is to return an ε-approximation of its maximum using the smallest possible number of function evaluations, the sample complexity of the algorithm. We provide an algorithm with sample complexity of order log^2(1/ε). This improves over previous results of Al-Mharmah and Calvin (1996) and Calvin et al. (2017) which provided only polynomial rates. Our algorithm is adaptive---each query depends on previous values---and is an instance of the optimism-in-the-face-of-uncertainty principle.

In physics and engineering literature, the distribution of the excursion-above-zero time distribution (exceedance distribution) for a stationary Gaussian process has been approximated by a stationary switching process with independently distributed switching times. The approach matched the covariance of the clipped Gaussian process with the one for the stationary switching process and the distribution of the latter was used as the so-called independent interval approximation (IIA). The approach successfully assessed the persistency exponent for many physically important processes but left an unanswered question when such an approach leads to a mathematically meaningful and proper exceedance distribution. Here we address this question by proposing an alternative matching of the expected values of the clipped Slepian process and the corresponding switched process initiated at the origin. The method has allowed resolving the mathematical correctness of the matching method for a large subclass of the Gaussian processes with monotonic covariance, for which we provide a sufficient condition for the validity of the IIA. Within this class, the IIA produces a valid distribution for the excursion time and is represented in an explicit stochastic form that connects directly to the covariance of the underlying Gaussian process. We compare the excursion level distributions as well as the corresponding persistency exponents obtained through the IIA method with numerically computed exact distributions, and the simulated distribution for several important Gaussian models. We also argue that for stationary Gaussian processes with a non-monotonic covariance, the IIA fails and should not be used.

Straight line equation y=mx with slope m, when singularly perturbed as ay^3+y=mx with a positive parameter a, results in S-shaped curves or S-curves on a real plane. As arightarrow 0, we get back y=mx which is a cumulative distribution function of a continuous uniform distribution that describes the occurrence of every event in an interval to be equally probable. As arightarrowinfty, the derivative of y has finite support only at y=0 resembling a degenerate distribution. Based on these arguments, in this work, we propose that these S-curves can represent maximum entropy uniform distribution to a zero entropy single value. We also argue that these S-curves are superposable as they are only parametrically nonlinear but fundamentally linear. So far, the superposed forms have been used to capture the patterns of natural systems such as nonlinear dynamics of biological growth and kinetics of enzyme reactions. Here, we attempt to use the S-curve and its superposed form as statistical models. We fit the models on a classical dataset containing flower measurements of iris plants and analyze their usefulness in pattern recognition. Based on these models, we claim that any non-uniform pattern can be represented as a singular perturbation to uniform distribution. However, our parametric estimation procedure have some limitations such as sensitivity to initial conditions depending on the data at hand.

Proper weight initialization prior to training has historically been one of the key factors that helped kick off the deep learning revolution. Initialization is even more crucial in "reservoir computing", where the weights of a readout layer are learned linearly while the reservoir weights are fixed and largely determine the richness, stability and memory of the resulting dynamics. In the infinite-width limit it has been shown that meaningful initializations are those sitting at an effective critical point of the randomly initialized model. The phase transition is controlled by the weight variance g^2 and separates an ordered phase from a chaotic one where information progressively degrades. Here we derive a simple criterion to estimate the critical g_c for a broad class of recurrent architectures and we show that it closely tracks the gain at which a gated-RNN reservoir achieves peak performance on a chaotic forecasting task. Finally, we argue that our criterion can serve as a design principle for future initialization schemes.

We explore the consequences of multi-trace deformations in applications of gauge-gravity duality to condensed matter physics. We find that they introduce a powerful new "knob" that can implement spontaneous symmetry breaking, and can be used to construct a new type of holographic superconductor. This knob can be tuned to drive the critical temperature to zero, leading to a new quantum critical point. We calculate nontrivial critical exponents, and show that fluctuations of the order parameter are `locally' quantum critical in the disordered phase. Most notably the dynamical critical exponent is determined by the dimension of an operator at the critical point. We argue that the results are robust against quantum corrections and discuss various generalizations.

Modern foundation models rely heavily on using scaling laws to guide crucial training decisions. Researchers often extrapolate the optimal architecture and hyper parameters settings from smaller training runs by describing the relationship between, loss, or task performance, and scale. All components of this process vary, from the specific equation being fit, to the training setup, to the optimization method. Each of these factors may affect the fitted law, and therefore, the conclusions of a given study. We discuss discrepancies in the conclusions that several prior works reach, on questions such as the optimal token to parameter ratio. We augment this discussion with our own analysis of the critical impact that changes in specific details may effect in a scaling study, and the resulting altered conclusions. Additionally, we survey over 50 papers that study scaling trends: while 45 of these papers quantify these trends using a power law, most under-report crucial details needed to reproduce their findings. To mitigate this, we we propose a checklist for authors to consider while contributing to scaling law research.

Kaplan et al. and Hoffmann et al. developed influential scaling laws for the optimal model size as a function of the compute budget, but these laws yield substantially different predictions. We explain the discrepancy by reproducing the Kaplan scaling law on two datasets (OpenWebText2 and RefinedWeb) and identifying three factors causing the difference: last layer computational cost, warmup duration, and scale-dependent optimizer tuning. With these factors corrected, we obtain excellent agreement with the Hoffmann et al. (i.e., "Chinchilla") scaling law. Counter to a hypothesis of Hoffmann et al., we find that careful learning rate decay is not essential for the validity of their scaling law. As a secondary result, we derive scaling laws for the optimal learning rate and batch size, finding that tuning the AdamW beta_2 parameter is essential at lower batch sizes.

Temperature scaling is a popular technique for tuning the sharpness of a model distribution. It is used extensively for sampling likely generations and calibrating model uncertainty, and even features as a controllable parameter to many large language models in deployment. However, autoregressive models rely on myopic temperature scaling that greedily optimizes the next token. To address this, we propose Long Horizon Temperature Scaling (LHTS), a novel approach for sampling from temperature-scaled joint distributions. LHTS is compatible with all likelihood-based models, and optimizes for the long-horizon likelihood of samples. We derive a temperature-dependent LHTS objective, and show that fine-tuning a model on a range of temperatures produces a single model capable of generation with a controllable long-horizon temperature parameter. We experiment with LHTS on image diffusion models and character/language autoregressive models, demonstrating advantages over myopic temperature scaling in likelihood and sample quality, and showing improvements in accuracy on a multiple choice analogy task by 10%.

We present an online post-hoc calibration method, called Online Platt Scaling (OPS), which combines the Platt scaling technique with online logistic regression. We demonstrate that OPS smoothly adapts between i.i.d. and non-i.i.d. settings with distribution drift. Further, in scenarios where the best Platt scaling model is itself miscalibrated, we enhance OPS by incorporating a recently developed technique called calibeating to make it more robust. Theoretically, our resulting OPS+calibeating method is guaranteed to be calibrated for adversarial outcome sequences. Empirically, it is effective on a range of synthetic and real-world datasets, with and without distribution drifts, achieving superior performance without hyperparameter tuning. Finally, we extend all OPS ideas to the beta scaling method.

Ecological systems are dynamic and policies to manage them need to respond to that variation. However, policy adjustments will sometimes be costly, which means that fine-tuning a policy to track variability in the environment very tightly will only sometimes be worthwhile. We use a classic fisheries management question -- how to manage a stochastically varying population using annually varying quotas in order to maximize profit -- to examine how costs of policy adjustment change optimal management recommendations. Costs of policy adjustment (here changes in fishing quotas through time) could take different forms. For example, these costs may respond to the size of the change being implemented, or there could be a fixed cost any time a quota change is made. We show how different forms of policy costs have contrasting implications for optimal policies. Though it is frequently assumed that costs to adjusting policies will dampen variation in the policy, we show that certain cost structures can actually increase variation through time. We further show that failing to account for adjustment costs has a consistently worse economic impact than would assuming these costs are present when they are not.

To stabilize the periodic operation of a chemical reactor the oscillation period should be determined precisely in real time. The method discussed in the paper is based on adaptive sampling of the state variable with the use of chaotic mapping to itself. It enables precise determination of the oscillation period in real time and could be used for a proper control system, that can successfully control the process and maintain the oscillation period at a set level. The method was applied to a tank reactor and tubular reactor with recycle.