Discovering Symbolic Differential Equations with Symmetry Invariants

TL;DR

This paper introduces a novel method for discovering symbolic differential equations by incorporating symmetry invariants, which ensures the equations respect physical laws and improves discovery accuracy and interpretability.

Contribution

The work proposes using symmetry invariants in equation discovery, integrating with existing methods to enhance their physical consistency and efficiency.

Findings

Successfully applied to fluid and reaction-diffusion systems

Recovered parsimonious, physically consistent equations

Improved accuracy and interpretability of discovered equations

Abstract

Discovering symbolic differential equations from data uncovers fundamental dynamical laws underlying complex systems. However, existing methods often struggle with the vast search space of equations and may produce equations that violate known physical laws. In this work, we address these problems by introducing the concept of symmetry invariants in equation discovery. We leverage the fact that differential equations admitting a symmetry group can be expressed in terms of differential invariants of symmetry transformations. Thus, we propose to use these invariants as atomic entities in equation discovery, ensuring the discovered equations satisfy the specified symmetry. Our approach integrates seamlessly with existing equation discovery methods such as sparse regression and genetic programming, improving their accuracy and efficiency. We validate the proposed method through applications to various physical systems, such as fluid and reaction-diffusion, demonstrating its ability to recover parsimonious and interpretable equations that respect the laws of physics.