Data-Driven Learning of Unknown Nonlinear Differential Equations Using Functional Analysis

Researchers propose a new interpretable machine learning method for data-driven discovery of nonlinear ordinary differential equations, learning unknown vector fields from a single state trajectory without prior physics knowledge. The method, based on functional analysis and operator theory, constructs a cost function in function space as an integral distance and uses an incremental learning algorithm for online updates, handling forced and unforced, autonomous and non-autonomous systems. Numerical examples demonstrate its advantages.

arXiv:2609.04329v1 Announce Type: new Abstract: 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.