Neural Symmetry Discovery: Learning Lie Algebra Generators from Variational Residuals

This paper presents a variational framework for the autonomous discovery of internal symmetry groups directly from observational field data. By extending the Neural Lagrangian framework to multi-component fields, we introduce a learnable Lie algebra generator into the optimization objective. We demonstrate that by minimizing the Euler-Lagrange residual alongside a novel symmetry-invariance loss, a neural network successfully identifies the SO(2) rotational symmetry of a complex scalar field system without any prior knowledge of the underlying group structure. The learned generator matrix converges to the canonical anti-symmetric form characteristic of so(2), confirming successful symmetry recovery. This approach provides a scalable path toward automated discovery of SU(2) and SU(3) gauge symmetries relevant to the Standard Model using purely data-driven variational methods.