从单条轨迹数据中解析未知非线性微分方程，无需物理先验。
Data-Driven Learning of Unknown Nonlinear Differential Equations Using Functional Analysis
- 基于泛函分析与算子理论构建函数空间损失，实现可解释建模。
- 仅需单条轨迹数据，即可同时识别系统动力学与外部作用力。
- 支持在线增量学习，适用于时变、非自治等复杂系统。
本文重新审视了基于数据发现非线性常微分方程（ODE）的问题，提出一种新的可解释机器学习方法。该方法旨在仅凭单一状态轨迹数据，不依赖系统物理先验知识，学习未知的向量场。与现有方法相比，该方法具有两大创新：1）其公式推导基于泛函分析与算子理论；2）损失函数在函数空间中定义为两个函数间积分距离，而非传统机器学习中的离散误差求和。进一步提出增量学习算法，支持在线处理新样本。该方法可同时发现受迫与无迫、自治与非自治（或时变）动力系统的未知向量场，并能同步识别随时间变化的外部力作为函数。最后通过数值实验验证了方法的优势。
原文摘要 · Abstract (English)
In this paper, the problem of data-driven discovery of nonlinear ordinary differential equations (ODEs) is recast, and a new interpretable machine learning (ML) method is proposed. The proposed method aims to learn the unknown vector field of nonlinear dynamics without prior knowledge of the system's physics from only one single state trajectory's data. The proposed method has two fundamental differences with existing methods: 1) the formulation presented in this method is derived based on Functional Analysis and Operator Theory, and 2) the cost function is constructed in the function space as a distance between two functions as an integral, instead of the discrete-sum of errors used in existing ML approaches. An incremental learning algorithm is proposed to learn the unknown vector field to handle new training samples in an online manner. The proposed method can discover the unknown vector field from both forced and unforced autonomous and non-autonomous (or time-varying) dynamical systems. The proposed method is able to simultaneously discover unknown external forces as a function of time and unknown underlying dynamics. Finally, numerical examples are given to demonstrate the advantages of the proposed method.
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