Data-Driven Discovery of Lie Algebra Generators via Fréchet-Invariant Learning

Ilya Markov, Alexander Hvatov

Keywords: neural operators, Lie algerbra, symmetry discovery, physics informed machine learning

Abstract: We present LieSym --- a framework for discovering infinitesimal generators of continuous symmetries directly from observational data without assuming prior knowledge of governing equations or symmetry groups. The method builds on Olver's geometric formalism and interprets symmetry discovery as the search for vector fields $v$ in the nullspace of Frechet derivative $DL[u]$, where $L$ is domain-specific operator (e.g., PDE residual or semantic classifier). To enforce structural consistency, we introduce Lie-orthonormal training objective that minimizes the Olver condition $v[Lop] = 0$ via automatic differentiation, enforces orthogonality of generators through stop-gradient regularisation, and ensures smoothness via Lipschitz constraints. We validate our approach on both physical and non-physical domains: for heat equation with periodic boundary conditions, we recover 3-dimensional symmetry subalgebra and quantify closure via Lie brackets $[v_i, v_j]$; for Oxford-IIIT Pet dataset, we discover semantic symmetries (pose adaptation, illumination invariance) and demonstrate their robustness through per-image validation. It is shown that spectrum of generator norms $|v_i|$ automatically reveals effective rank of symmetry algebra --- capability that is absent in prior work. The proposed method allows to bridge operator learning, Lie theory, and semantic AI, enabling equation-free discovery of structured invariances.

Submission Number: 61