发表机构
University of Minnesota-Twin Cities; University of Utah; National Institute of Standards and Technology（明尼苏达大学双城分校; 犹他大学; 美国国家标准与技术研究院）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
该研究提出一种基于泛函分析和算子理论的可解释机器学习方法，可从单一状态轨迹数据中学习非线性动力学的未知向量场，还能同时发现未知外力，通过数值示例验证了其优势。
AI 中文摘要
本文重新阐述了非线性常微分方程（ODE）的数据驱动发现问题，并提出了一种新的可解释机器学习（ML）方法。该方法旨在仅从单一状态轨迹的数据中学习非线性动力学的未知向量场，无需预先了解系统的物理特性。该方法与现有方法有两个根本差异：1）其公式基于泛函分析和算子理论推导得出；2）代价函数在函数空间中构建为两个函数之间的积分距离，而非现有ML方法采用的误差离散和。文中提出了一种增量学习算法，用于在线处理新训练样本以学习未知向量场。该方法可从受迫和未受迫的自治、非自治（或时变）动力学系统中发现未知向量场，还能同时发现作为时间函数的未知外力和未知基础动力学。最后，通过数值示例证明了该方法的优势。
英文摘要
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.
