这是PDE发现评估的第一篇系统综述，适合ML和物理科学交叉领域的研究者。它帮你理清预测精度与物理一致性之间的权衡，避免用单一指标误导结论。
这篇论文针对偏微分方程（PDE）发现领域的事后评估问题，提出了首个PDE评估指标分类法。作者系统梳理了来自机器学习和数值分析的文献，指出评估需同时考虑预测准确性、物理一致性、可解释性和分布外泛化能力等四个冲突维度。现有指标仅部分覆盖整体问题，可能导致对物理理论有效性的过度解读。论文提供了标准化实践建议，旨在推动可靠评估方法的发展。
On the post-hoc Evaluation of PDE Discovery: A Multifaceted Challenge of Scientific Advancement
Partial differential equation (PDE) discovery aims to identify from data the governing law of a physical system. Constituting a cornerstone of scientific advancement, it has become during the past decade a major line of research in the rapidly evolving field of Physics-informed Machine Learning (PiML). Among the remaining open problems to address in this domain, the post-hoc evaluation of discovered PDEs raises the particular difficulty of being multifaceted. Indeed, it requires jointly considering predictive accuracy, physical consistency, interpretability, and out-of-distribution generalization capacity. Given that some of these properties are conflicting, it is worth noting that the wide range of existing evaluation metrics only partially address the overall problem, potentially leading to overly interpreted conclusions about the validity of a presumed new physical theory. From an abundant literature spanning machine learning, numerical analysis, information theory or symbolic regression, we propose, to our knowledge, the first taxonomy of PDE evaluation metrics, and discuss their advantages and limitations in depth. Based on the observation that evaluation is often achieved on a case-by-case basis and that a universally accepted methodology remains elusive, we further provide recommendations with the aim of promoting standardized and reliable practices, before sketching promising future lines of research in this field. We argue that this paper is intended both for ML experts who design new PDE discovery algorithms and for users of these methods aiming, in real applications, to discover and validate well-founded scientific laws.