Discovering Symmetries of ODEs by Symbolic Regression

TL;DR

This paper introduces a symbolic regression method to discover Lie point symmetries of ODEs, enabling the identification of symmetries that traditional computer algebra systems often miss, thus aiding in solving complex nonlinear differential equations.

Contribution

It adapts search-based symbolic regression to find generators of Lie point symmetries, improving symmetry detection beyond existing CAS capabilities.

Findings

Successfully finds symmetries missed by CASs

Enhances automated ODE solving techniques

Demonstrates effectiveness on nonlinear systems

Abstract

Solving systems of ordinary differential equations (ODEs) is essential when it comes to understanding the behavior of dynamical systems. Yet, automated solving remains challenging, in particular for nonlinear systems. Computer algebra systems (CASs) provide support for solving ODEs by first simplifying them, in particular through the use of Lie point symmetries. Finding these symmetries is, however, itself a difficult problem for CASs. Recent works in symbolic regression have shown promising results for recovering symbolic expressions from data. Here, we adapt search-based symbolic regression to the task of finding generators of Lie point symmetries. With this approach, we can find symmetries of ODEs that existing CASs cannot find.