D-CHOPT: DISCOVERING CLOSED-FORM HIGH-DIMENSIONAL ODEs FROM PARTIAL OBSERVED TRAJECTORIES

Yiting Duan, Zhao Yao, Yi Guo

Keywords: Ordinary Differential Equation, Sparse Identification of Nonlinear Dynamic System, Partial Observed Data

TL;DR: In this work, we propose a method for discovering closed-form high-dimensional ODEs from partially observed trajectories, called D-CHOT, which advances ODE discovery methods beyond the natural limitations of high-dimensional ODEs.

Abstract: Machine learning algorithms have become a new paradigm for automatically discovering closed-form ordinary differential equations (ODEs) from observed trajectories. Although significant breakthroughs have been made in this field, such as symbolic regression and sparse identification of nonlinear dynamics (SINDy), existing approaches primarily perform well for low-dimensional ODEs. This limitation arises due to the lack of understanding of observability limitations in partially observed trajectories, and the additional challenges introduced by complex topological properties. In this work, we propose a method for discovering closed-form high-dimensional ODEs from partially observed trajectories, called D-CHOPT, which advances ODE discovery methods beyond the natural limitations of high-dimensional ODEs. D-CHOPT uses an invertible neural network as the backbone to find the optimal solution within the diffeomorphic equivariant group of the reconstructed dynamical systems, while preserving topological properties and integrating a variable selection method. We provide a formal analysis of observability and the learning limitations of partial trajectories, and explain the enhancements in a manner consistent with the theoretical results. In experiments, D-CHOPT successfully discovered the governing equations for a wide range of dynamical systems, both low and high dimensional.

Primary Area: learning on time series and dynamical systems

Submission Number: 8205