神经偏微分方程求解器是否学习了正确的动力学？

中文摘要 AI 辅助

神经偏微分方程求解器能够实现较低的预测误差，但它们是否再现了所建模系统的动力学？仅凭预测分数无法给出完整答案：它们衡量与参考解的一致性，但对误差如何累积、邻近状态如何发散或极端事件如何产生提供的洞察有限。我们提出了一个评估框架，直接检验确定性和随机神经求解器中的这些行为。通过演化邻近初始状态的集合，并与直接数值模拟进行比较，我们评估了学习动力学的三个互补方面：误差形成、集合几何和极端事件。在二维柯尔莫哥洛夫流上的实验揭示了传统分数可能掩盖的局限性。较小的轨迹误差可能反映较弱的误差放大，尽管局部更新不太准确。模型可以匹配集合的整体扩散和有效维度，但未能捕捉邻近状态发散的空间方向。同样，匹配整体事件频率可能掩盖无法预测持续性极端事件的问题。这些发现表明，改进的预测精度并不必然意味着更高的动力学保真度。我们的框架使这种区别可测量，为评估神经偏微分方程求解器的进展是否更好地捕捉底层动力学提供了具体标准。

英文摘要

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.

发表机构

• Center for Combustion Energy, Tsinghua University（清华大学燃烧能源中心）
• College of AI, Tsinghua University（清华大学人工智能学院）
• Department of Mechanical Engineering, University College London（伦敦大学学院机械工程系）

机构由 AI 辅助整理，请以论文原文为准。