用遍历理论评估时序数据学出的动态系统是否真实反映物理规律
When are dynamical systems learned from time series data statistically accurate?
- 基于遍历理论定义动态系统学习的泛化，关注不变测度而非仅预测误差
- 发现仅回归的神经微分方程在统计上不准确，加入雅可比信息后显著改善
- 适用于混沌系统，对MLP、ResNet、RNN等模型均验证有效
传统泛化概念常无法描述从动态数据中学习的模型捕捉有意义信息的能力。一个测试误差小的神经网络可能仍无法复现系统的物理行为，包括统计矩和李雅普诺夫指数。为填补这一空白，我们提出一种基于遍历理论的泛化方法，用于评估从时间序列数据中学习的复杂动态模型。主要贡献在于定义并分析了多种神经表示在广义遍历系统（包括混沌系统）上的泛化能力，能有效模拟底层不变物理测度。结果为生成类动态系统（如神经微分方程）的回归方法为何泛化失败提供了理论解释，并说明在训练中加入雅可比信息可提升其统计准确性。我们在多个遍历混沌系统及神经网络结构（包括MLP、ResNet、傅里叶神经层、RNN）上验证了结论。
原文摘要 · Abstract (English)
Conventional notions of generalization often fail to describe the ability of learned models to capture meaningful information from dynamical data. A neural network that learns complex dynamics with a small test error may still fail to reproduce its \emph{physical} behavior, including associated statistical moments and Lyapunov exponents. To address this gap, we propose an ergodic theoretic approach to generalization of complex dynamical models learned from time series data. Our main contribution is to define and analyze generalization of a broad suite of neural representations of classes of ergodic systems, including chaotic systems, in a way that captures emulating underlying invariant, physical measures. Our results provide theoretical justification for why regression methods for generators of dynamical systems (Neural ODEs) fail to generalize, and why their statistical accuracy improves upon adding Jacobian information during training. We verify our results on a number of ergodic chaotic systems and neural network parameterizations, including MLPs, ResNets, Fourier Neural layers, and RNNs.
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