Discovering Symmetries of ODEs by Symbolic Regression

2025-06-24

AI Summary

This work addresses the challenge of automatically identifying Lie point symmetries for nonlinear ordinary differential equations (ODEs). Unlike conventional computer algebra systems—which rely on manual ansatz construction or suffer from algebraic complexity—we propose a novel, fully automated method based on expression-search symbolic regression, tightly integrated with Lie group theory. Our approach directly infers symmetry generators from the ODE’s structural form without imposing prior functional assumptions, enabling systematic exploration of the function space. It successfully discovers nontrivial symmetries undetected by state-of-the-art systems such as Mathematica and Maple. Experimental results demonstrate that the identified symmetries effectively reduce ODE order or yield conserved quantities, thereby enhancing both the automation and interpretability of nonlinear ODE solving. This represents the first integration of symbolic regression with Lie symmetry analysis for autonomous symmetry discovery.

Technology Category

• Search and Optimization: Mixed Discrete/Continuous Search
• Constraint Satisfaction and Optimization: Satisfiability Modulo Theories
• Machine Learning: Neuro-Symbolic Learning

Application Category

• Graph Algorithms and Modeling for the Web: Representation, reconstruction, and subgraph or motif discovery in Web-related graphs
• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking
• Systems and Infrastructure for Web, Mobile and WoT: Applied ML and AI for Web-based mobile applications

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.

Problem

• Finding Lie point symmetries for ODEs automatically
• Improving symbolic regression for symmetry detection
• Enhancing CAS capabilities for nonlinear ODE solving

Innovation

• Symbolic regression for Lie symmetries
• Search-based discovery of ODE symmetries
• Enhancing CAS capabilities with AI