AI 中文总结
本文针对偏微分方程发现的事后评估这一多方面挑战，提出首个评估指标分类法，深入探讨优缺点并给出建议，旨在为设计新算法的专家及实际应用中发现和验证科学定律的用户提供参考。
AI 中文摘要
偏微分方程（PDE）发现旨在从数据中识别物理系统的 governing law。它是科学进步的基石，在过去十年成为物理信息机器学习（PiML）快速发展领域的主要研究方向。该领域现存的开放问题中，已发现 PDE 的事后评估存在多方面特殊困难，需要综合考虑预测准确性、物理一致性、可解释性和分布外泛化能力等。现有评估指标只能部分解决整体问题。本文提出了首个 PDE 评估指标分类法，深入讨论其优缺点，并给出促进标准化和可靠实践的建议。
英文摘要
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.
