Discovering Symbolic Differential Equations with Symmetry Invariants
This work addresses the challenge of discovering physically consistent equations from data, which is important for researchers in physics and engineering, though it is incremental as it builds on existing methods like sparse regression and genetic programming.
The authors tackled the problem of discovering symbolic differential equations from data by introducing symmetry invariants to ensure equations respect physical laws, resulting in improved accuracy and efficiency in recovering parsimonious and interpretable equations for systems like fluid and reaction-diffusion.
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 fluid and reaction-diffusion, demonstrating its ability to recover parsimonious and interpretable equations that respect the laws of physics.