Method finds continuous symmetries directly from differential data
Data-driven discovery of intrinsic Lie point symmetries of differential equation systems
Machine Learning
Summary
Finding hidden symmetries in equations that describe systems can help simplify and better understand those systems. This paper introduces a method that discovers such symmetries directly from sampled data without needing the exact formulas. The authors test their approach on a known benchmark and recover all expected symmetries and their relationships accurately. This means the method can reveal important geometric features of systems just from data.
What this means in practice
- •For computational physicists: Identify intrinsic symmetries from simulated or measured data to simplify complex physical models without symbolic manipulation.
- •For machine learning engineers: Extract continuous symmetries from data to improve model reduction and scientific machine learning architectures.
Authors
Ilker Gurcan, Kadir Ustun, Ramazan Gokberk Cinbis
Abstract
Discovering continuous symmetries directly from differential-equation data can reveal geometric structure for model reduction and scientific machine learning without symbolic determining equations or a closed-form representation of the governing system. We formulate intrinsic Lie point-symmetry discovery as an operator-theoretic problem on sampled jet-space data. Admissible generators are identified from a sampled determining operator and separated from redundant, solution-dependent, and geometrically trivial directions. The framework combines prolongation, sequential subspace restrictions, and a fixed nonlinear dictionary linear in trainable coefficients. A rank-stabilization preflight test diagnoses systems whose represented intrinsic symmetry algebra does not stabilize under dictionary refinement. On a controlled differential-equation benchmark with an analytically known 11-dimensional represented intrinsic symmetry algebra, the method recovers the complete represented generator set, reproduces its Lie-bracket structure to numerical precision, and approximates a known generator beyond the training domain. Here we show that explicit intrinsic continuous-symmetry generators can be recovered from data obtained from exact or approximate differential systems.