Error-Conditioned Neural Solvers (ENS) improve PDE surrogate accuracy by feeding residual fields as inputs
A new arXiv preprint introduces Error-Conditioned Neural Solvers (ENS), a method for neural PDE surrogates that passes the PDE residual field directly as input to the network at each iteration, enabling iterative self-correction rather than gradient-based residual minimization. The authors demonstrate theoretically and empirically that residual minimization is an unreliable proxy for reconstruction accuracy in ill-conditioned systems, explaining failures of prior hybrid methods. ENS achieves up to 10× accuracy gains on turbulent Kolmogorov flow across four PDE families, with lower compute cost than hybrid approaches and generalization under distribution shift including zero-shot cross-equation transfer.
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Physics-informed Fourier-wavelet transformer improves multiscale CFD surrogate modeling
A new arXiv preprint introduces a physics-informed Fourier-wavelet transformer for next-step velocity-field reconstruction in computational fluid dynamics, combining hybrid spectral encoding with PDE-residual-guided self-attention and self-supervised pretraining. The model is evaluated on cylinder-wake and fluid-structure interaction benchmarks, achieving best-in-class normalized mean-squared error on both tasks and stronger recovery of localized flow structures compared to spectral, transformer, and physics-informed neural network baselines. The work targets the persistent gap between global flow pattern accuracy and fine-grained multiscale structure recovery in surrogate models.
Exact Posterior Score (EPS): Closed-form posterior sampling for linear inverse problems with diffusion models
A new arXiv preprint derives the exact posterior score in closed form for linear Gaussian inverse problems under general Gaussian interpolants, showing that posterior sampling reduces to a denoising problem at an operator-dependent shifted pivot under anisotropic noise covariance. The authors convert this identity into a training objective called Exact Posterior Score (EPS) that preserves the input/output structure of standard diffusion pretraining, enabling training from scratch or fine-tuning from a pretrained denoiser. EPS is evaluated on five linear inverse problems across FFHQ and ImageNet, outperforming both training-free and training-based baselines while requiring roughly an order of magnitude fewer denoiser evaluations than gradient-based posterior samplers.
Tight Convergence Theory for Error Feedback Algorithms in Distributed Optimization
This paper provides tight convergence analyses for two major error-feedback algorithms—classic Error Feedback (EF) and Error Feedback 21 (EF21)—used to mitigate communication bottlenecks in distributed learning. The authors identify optimal step-size choices and construct tailored Lyapunov functions for each method, yielding guarantees that hold independently of the number of agents and recover the best known single-agent bounds. The work clarifies the relative performance of these gradient compression variants, which has remained poorly understood despite widespread use.
Equilibrium Reasoners: Learning Attractors Enables Scalable Reasoning
This paper introduces Equilibrium Reasoners (EqR), a framework that formalizes test-time compute scaling through learned task-conditioned attractors in latent space, where stable fixed points correspond to valid solutions. EqR scales along two axes—depth (more iterations) and breadth (aggregating stochastic trajectories)—without requiring external verifiers or task-specific priors. On Sudoku-Extreme, unrolling up to 40,000 equivalent layers boosts accuracy from 2.6% (feedforward baseline) to over 99%. The work provides a mechanistic lens for understanding why iterative latent models generalize beyond memorized patterns.
SURGE: Approximation-free Training-Free Particle Filter for Diffusion Surrogate
The paper introduces URGE (Unbiased Resampling via Girsanov Estimation), a derivative-free inference-time scaling algorithm for diffusion models that performs path-wise importance reweighting using a Girsanov change of measure. Unlike existing inference-time guidance methods, URGE requires no score, Hessian, or PDE evaluations, attaching multiplicative weights to simulated trajectories and periodically resampling. The authors establish a theoretical equivalence between path-wise and particle-wise sequential Monte Carlo (SMC), guaranteeing unbiased terminal distributions. Empirically, URGE outperforms existing inference-time guidance baselines on synthetic tests and diffusion-model benchmarks while being simpler to implement.
SAERL: Using Sparse Autoencoders to Guide LLM Reinforcement Learning Data Engineering
SAERL is a post-training data engineering framework that uses Sparse Autoencoders (SAEs) — a mechanistic interpretability tool — to extract intrinsic model signals for controlling data diversity, difficulty, and quality during RL fine-tuning. The framework applies SAE-space clustering for batch diversity, a difficulty proxy for curriculum ordering, and a quality probe for data filtering. On Qwen2.5-Math-1.5B with GRPO, SAERL achieves 3% average accuracy improvement and reaches target accuracy with 20% fewer training steps. SAE representations transfer across model families and scales, suggesting broad applicability as a lightweight data engineering tool.
Gradient Equilibrium shown equivalent to Blackwell Approachability in online learning
A new arXiv preprint proves that gradient equilibrium (GEQ), a recently introduced online optimization framework generalizing first-order stationarity, is algorithmically equivalent to Blackwell approachability. The equivalence implies GEQ is also equivalent to regret minimization and calibration, resolving an open question about GEQ's place in the online learning landscape. The reductions are efficient and allow transfer of refined guarantees like optimism and strong adaptivity from regret minimization to GEQ, with applications including online conformal prediction.
Solver-dependent Nash equilibrium selection on zero-sum polytopes: regularized methods select max-entropy members
A new arXiv preprint investigates whether different Nash equilibrium solvers systematically select different members of the Nash polytope in two-player zero-sum games. Using six analytically tractable games including Kuhn poker, the authors find that regularized last-iterate methods (R-NaD, magnetic mirror descent) converge to the maximum-entropy Nash equilibrium — interpretable as an information projection — while regret-averaging methods (CFR, CFR+, fictitious play) drift to lower-entropy boundary solutions. The distinction has downstream consequences for performance against sub-optimal opponents in games with sequential or hidden-information structure, with implications for multi-agent AI training and game-solving pipelines.