Title: Governing Equation Discovery from Data Based on Differential Invariants
Primary area: learning on time series and dynamical systems
Keywords: equation discovery, symmetry, Lie theory, equivariant neural network

Abstract: The explicit governing equation is one of the simplest and most intuitive forms for characterizing physical laws. However, directly discovering partial differential equations (PDEs) from data poses significant challenges, primarily in determining relevant terms from a vast search space. Symmetry, as a crucial prior knowledge in scientific fields, has been widely applied in tasks such as designing equivariant networks and guiding neural PDE solvers. In this paper, we propose a pipeline for governing equation discovery based on differential invariants, which can losslessly reduce the search space of existing equation discovery methods while strictly adhering to symmetry. Specifically, we compute the set of differential invariants corresponding to the infinitesimal generators of the symmetry group and select them as the relevant terms for equation discovery. Taking DI-SINDy (SINDy based on Differential Invariants) as an example, we demonstrate that its success rate and accuracy in PDE discovery surpass those of other symmetry-informed governing equation discovery methods across a series of PDEs. Additional results further indicate that our method exhibits strong robustness to data and symmetry noise, as well as significant potential for solving high-dimensional dynamic systems.

TL;DR: Symmetry-guided equation discovery based on differential invariants