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This research advances interpretable machine learning by leveraging functional analysis to discover nonlinear ordinary differential equations from limited trajectory data without prior physical knowledge. The method distinguishes itself through an integral-based cost function in function space and an incremental learning algorithm capable of handling online updates for both autonomous and non-autonomous systems. These developments offer a robust framework for modeling complex dynamical systems where underlying physics remains unknown.
Data-Driven Learning of Unknown Nonlinear Differential Equations Using Functional Analysis
arXiv:2609.04329v1 Announce Type: cross 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.
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.
Machine Learning (cs.LG); Artificial Intelligence (cs.AI); Systems and Control (eess.SY)
arXiv:2609.04329 [cs.LG]
(or arXiv:2609.04329v1 [cs.LG] for this version)
https://doi.org/10.48550/arXiv.2609.04329 Focus to learn more arXiv-issued DOI via DataCite (pending registration)
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