Data-driven Lie point symmetry detection for continuous dynamical systems

Abstract

Symmetry detection, the task of discovering the underlying symmetries of a given dataset, has been gaining popularity in the machine learning community, particularly in science and engineering applications. Most previous works focus on detecting ‘canonical’ symmetries such as translation, scaling, and rotation, and cast the task as a modeling problem involving complex inductive biases and architecture design of neural networks. We challenge these assumptions and propose that instead of constructing biases, we can learn to detect symmetries from raw data without prior knowledge. The approach presented in this paper provides a flexible way to scale up the detection procedure to non-canonical symmetries, and has the potential to detect both known and unknown symmetries alike. Concretely, we focus on predicting the generators of Lie point symmetries of partial differential equations, more specifically, evolutionary equations for ease of data generation. Our results demonstrate that well-established neural network architectures are capable of recognizing symmetry generators, even in unseen dynamical systems. These findings have the potential to make non-canonical symmetries more accessible to applications, including model selection, sparse identification, and data interpretability.

Keywords

• differential equations
• symmetry detection
• Lie theory