Title: Discovering Symbolic Differential Equations with Symmetry Invariants
Primary area: applications to physical sciences (physics, chemistry, biology, etc.)
Keywords: equation discovery, symbolic regression, symmetry, PDE, Lie point symmetry, equivariance

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 \textit{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 Darcy flow and reaction-diffusion, demonstrating its ability to recover parsimonious and interpretable equations that respect the laws of physics.

TL;DR: Enforcing symmetry via the use of differential invariants in symbolic differential equation discovery algorithms to improve their accuracy and efficiency.