In science we are interested in finding the governing equations, the dynamical rules, underlying empirical phenomena. While traditionally scientific models are derived through cycles of human insight and experimentation, recently deep learning (DL) techniques have been advanced to reconstruct dynamical systems (DS) directly from time series data. State-of-the-art dynamical systems reconstruction (DSR) methods show promise in capturing invariant and long-term properties of observed DS, but their ability to generalize to unobserved domains remains an open challenge. Yet, this is a crucial property we would expect from any viable scientific theory. In this work, we provide a formal framework that addresses generalization in DSR. We explain why and how out-of-domain (OOD) generalization (OODG) in DSR profoundly differs from OODG considered elsewhere in machine learning. We introduce mathematical notions based on topological concepts and ergodic theory to formalize the idea of learnability of a DSR model. We formally prove that black-box DL techniques, without adequate structural priors, generally will not be able to learn a generalizing DSR model. We also show this empirically, considering major classes of DSR algorithms proposed so far, and illustrate where and why they fail to generalize across the whole phase space. Our study provides the first comprehensive mathematical treatment of OODG in DSR, and gives a deeper conceptual understanding of where the fundamental problems in OODG lie and how they could possibly be addressed in practice.

翻译：在科学领域，我们致力于发现经验现象背后的控制方程与动力学规律。传统科学模型通常通过人类洞察与实验的循环推演而建立，而近年来深度学习技术已取得进展，能够直接从时间序列数据中重建动态系统。最先进的动态系统重建方法在捕捉观测动态系统的不变特性与长期属性方面展现出潜力，但其在未观测域中的泛化能力仍是待解难题——然而这恰恰是任何可行科学理论应具备的关键特性。本研究提出一个形式化框架来探讨动态系统重建中的泛化问题，阐释域外泛化在动态系统重建中为何及如何与机器学习其他领域的域外泛化存在本质区别。我们基于拓扑学概念与遍历理论构建数学工具，形式化定义了动态系统重建模型的可学习性，并严格证明：缺乏适当结构先验的黑盒深度学习技术通常无法学习到具备泛化能力的动态系统重建模型。通过评估现有主要类别的动态系统重建算法，我们进行实验验证，展示了这些算法为何及何处难以实现全相空间的泛化。本研究首次对动态系统重建中的域外泛化进行了系统性的数学阐释，为理解域外泛化的根本问题所在及其可能的实践解决方案提供了更深层的概念框架。