Structure-preserving fidelity diagnostics for partial differential equations

Kyriakos Flouris

Home/Archive/arXiv

arXiv · 2610.17483

Structure-preserving fidelity diagnostics for partial differential equations

Kyriakos Flouris

View original source ↗Read source document ↗Export metadata ↓

Abstract

Machine learning surrogates of partial differential equations (PDEs) are commonly trained on data generated by numerical solvers. Therefore, the training data obey not the PDE but the modified equation of the numerical scheme, and the test loss, computed on data from the same solver, cannot distinguish a model that learned the physics from one that learned the discretization error. Here, we show that the difference is nevertheless visible in the trained model. We propose a suite of structure-preserving fidelity diagnostics, simple scores that measure whether a learned model respects one known property of the physics, such as a conserved quantity, a symmetry or a Hamiltonian structure, without access to the training data or the governing equation. Over five systems, the diagnostics track the error with respect to the true dynamics with a rank correlation of up to $0.94$, while the test loss is uncorrelated or anti-correlated with it. For the Burgers, Korteweg-de Vries and Navier-Stokes equations, the diagnostics separate numerical schemes that the test loss ranks as equivalent, including schemes of the same order. order.

Explore related subjects

cs.LG physics.comp-ph

Keep this discovery

Explore connections, maps & timelines

BibTeX RIS

Kyriakos Flouris. 2026-10-12. Structure-preserving fidelity diagnostics for partial differential equations. https://arxiv.org/abs/2610.17483

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

Model Predictive Task Sampling for Efficient and Robust Adaptation

Foundation models have revolutionized general-purpose problem-solving, offering rapid task adaptation through pretraining, meta-training, and finetuning. Recent crucial advances in these paradigms reveal the importance of challenging task prioritized sampling to enhance adaptation robustness under distribution shifts. However, ranking task difficulties over iteration as a preliminary step typically requires exhaustive task evaluation, which is practically unaffordable in computation and data-annotation. This study provides a novel perspective to illuminate the possibility of leveraging the dual importance of adaptation robustness and learning efficiency, particularly in scenarios where task evaluation is risky or costly, such as iterative agent-environment interactions for robotic policy evaluation or computationally intensive inference steps for finetuning foundation models. Firstly, we introduce Model Predictive Task Sampling (MPTS), a framework that bridges the task space and adaptation risk distributions, providing a theoretical foundation for robust active task sampling. MPTS employs a generative model to characterize the episodic optimization process and predicts task-specific adaptation risk via posterior inference. The resulting risk predictive model amortizes the costly evaluation of task adaptation performance and provably approximates task difficulty rankings. MPTS seamlessly integrates into zero-shot, few-shot, and supervised finetuning settings. Empirically, we conduct extensive experiments in pattern recognition using foundation models and sequential decision-making. Our results demonstrate that MPTS significantly enhances adaptation robustness for tail risk or out-of-distribution (OOD) tasks and improves learning efficiency compared to state-of-the-art (SoTA) methods. The code is available at the project site https://github.com/thu-rllab/MPTS.

cs.LG↗

arXiv2026

Asynchronous Sharpness-Aware Minimization For Fast and Accurate Deep Learning

Sharpness-Aware Minimization (SAM) is an optimization method that improves the generalization performance of machine learning models. Despite its effectiveness, it has not been widely adopted in real-world applications due to its high computational cost. Existing approaches reduce this overhead by approximating or reusing gradients in the perturbation step, but still assume sequential perturbation and update steps. In this study, we introduce AsyncSAM, a variant of SAM that enables concurrent model perturbation and parameter updates across multiple compute resources. By adjusting the perturbation batch size in a system-aware manner, our asynchronous scheme fully overlaps perturbation with updates, eliminating the overhead and allowing even CPUs to perform perturbations without extra time. We also provide theoretical analysis showing that AsyncSAM achieves the same order of convergence as conventional SAM. Extensive experiments demonstrate that AsyncSAM delivers superior generalization performance and the highest training throughput among state-of-the-art SAM variants. For example, in ViT fine-tuning on CIFAR-100, it achieves accuracy comparable to SAM while maintaining training time nearly identical to SGD.

cs.LG↗

arXiv2026

Weisfeiler and Leman Follow the Arrow of Time: Expressive Power of Message Passing in Temporal Event Graphs

An important characteristic of temporal graphs is how the directed arrow of time influences their causal topology, i.e., which nodes can possibly influence each other causally via time-respecting paths. The resulting patterns are often neglected by temporal graph neural networks (TGNNs). To formally analyze the expressive power of TGNNs, we lack a generalization of graph isomorphism to temporal graphs that fully captures their causal topology. Addressing this gap, we introduce the notion of consistent event graph isomorphism, which utilizes a time-unfolded representation of time-respecting paths in temporal graphs. We compare this definition with existing notions of temporal graph isomorphisms. We illustrate and highlight the advantages of our approach and develop a temporal generalization of the Weisfeiler-Leman algorithm to heuristically distinguish non-isomorphic temporal graphs. Building on this theoretical foundation, we derive a novel message passing scheme for temporal graph neural networks that operates on the event graph representation of temporal graphs. An experimental evaluation shows that our approach performs well in a temporal graph classification experiment.

cs.LG↗