A model-free method for discovering symmetry in differential equations

TL;DR

This paper presents a data-driven, model-free numerical method to discover symmetries in differential equations by approximating infinitesimal generators directly from scattered data on unknown manifolds.

Contribution

It introduces a novel numerical scheme using manifold learning to identify continuous symmetries without explicit differential equation forms.

Findings

Accurately recovers symmetries from scattered data.

Demonstrates robustness and convergence in numerical experiments.

Applicable to both ordinary and partial differential equations.

Abstract

Symmetry in differential equations reveals invariances and offers a powerful means to reduce model complexity. Lie group analysis characterizes these symmetries through infinitesimal generators, which provide a local, linear criterion for invariance. However, identifying Lie symmetries directly from scattered data, without explicit knowledge of the governing equations, remains a significant challenge. This work introduces a numerical scheme that approximates infinitesimal generators from data sampled on an unknown smooth manifold, enabling the recovery of continuous symmetries without requiring the analytical form of the differential equations. We employ a manifold learning technique, Generalized Moving Least Squares, to prolongate the data, from which a linear system is constructed whose null space encodes the infinitesimal generators representing the symmetries. Convergence bounds for the proposed approach are derived. Several numerical experiments, including ordinary and partial differential equations, demonstrate the method's accuracy, robustness, and convergence, highlighting its potential for data-driven discovery of symmetries in dynamical systems.