Learning Governing Equations of Unobserved States in Dynamical Systems

TL;DR

This paper introduces a hybrid neural ODE and symbolic regression approach to learn governing equations of unobserved states in partially-observed dynamical systems, demonstrating success on ecological and chaotic models with noise robustness.

Contribution

It presents a novel combination of neural ODEs and symbolic regression for modeling unobserved states in dynamical systems, addressing partial observability.

Findings

Successfully learned true governing equations of unobserved states

Demonstrated robustness to measurement noise

Applied to Lotka-Volterra and Lorenz systems

Abstract

Data-driven modelling and scientific machine learning have been responsible for significant advances in determining suitable models to describe data. Within dynamical systems, neural ordinary differential equations (ODEs), where the system equations are set to be governed by a neural network, have become a popular tool for this challenge in recent years. However, less emphasis has been placed on systems that are only partially-observed. In this work, we employ a hybrid neural ODE structure, where the system equations are governed by a combination of a neural network and domain-specific knowledge, together with symbolic regression (SR), to learn governing equations of partially-observed dynamical systems. We test this approach on two case studies: A 3-dimensional model of the Lotka-Volterra system and a 5-dimensional model of the Lorenz system. We demonstrate that the method is capable of successfully learning the true underlying governing equations of unobserved states within these systems, with robustness to measurement noise.