Do Neural PDE Solvers Learn the Right Dynamics?

2026-10-03

AI Summary

This study addresses the ambiguity in the dynamical fidelity of neural PDE solvers, which exhibit low prediction errors yet whose traditional evaluation metrics fail to reveal dynamic deficiencies such as error accumulation and extreme event misprediction. To this end, we propose a multidimensional evaluation framework that evolves ensembles of neighboring initial states and benchmarks them against direct numerical simulations, systematically examining the dynamical characteristics of deterministic and stochastic solvers through error formation, ensemble geometry, and extreme events. This work establishes new criteria distinguishing predictive accuracy from dynamical fidelity, demonstrating that small trajectory errors may stem from weak error amplification and that matched statistics can mask spatial divergence and failures in extreme event prediction. Furthermore, it verifies that improving accuracy does not necessarily enhance fidelity, providing measurable criteria for assessing whether models genuinely capture underlying physical dynamics.

Abstract

Neural PDE solvers can achieve low prediction errors, but do they reproduce the dynamics of the systems they model? Prediction scores alone offer an incomplete answer: they measure agreement with reference solutions but provide limited insight into how errors accumulate, nearby states diverge, or extreme events arise. We propose an evaluation framework that directly examines these behaviors in deterministic and stochastic neural solvers. By evolving ensembles of nearby initial states and comparing them with direct numerical simulation, we assess three complementary aspects of learned dynamics: error formation, ensemble geometry, and extreme events. Experiments on two-dimensional Kolmogorov flow reveal limitations that conventional scores can obscure. Smaller trajectory errors can reflect weaker error amplification despite less accurate local updates. Models can match an ensemble's overall spread and effective dimension while failing to capture the spatial directions where nearby states diverge. Similarly, matching overall event frequencies can conceal failures to predict persistent extreme events. These findings show that improved prediction accuracy does not necessarily imply greater dynamical fidelity. Our framework makes this distinction measurable, providing concrete criteria for evaluating whether advances in neural PDE solvers better capture the underlying dynamics.

Problem

• Neural PDE solvers
• dynamical fidelity
• error accumulation
• extreme events
• evaluation framework

Innovation

• Neural PDE Solvers
• Dynamics Fidelity
• Evaluation Framework
• Ensemble Geometry
• Extreme Events