A Stability Theory for Generator-Based Continuous Symmetry Learning

Symmetry discovery learns transformations that preserve functions, data distributions, or the solution sets of differential equations, revealing structure that can guide model design and scientific understanding. Existing methods address these target objects through different formulations, making it difficult to identify common conditions for success. We develop a stability theory for generator-based symmetry discovery by formulating it as a subspace recovery problem through a task-specific symmetry-measuring operator. This provides a common framework for analyzing how surrogate models and numerical approximations affect recovered symmetries. We establish sufficient conditions for accurate recovery and derive an error bound governed by the surrogate operator error relative to the separation between symmetry and non-symmetry directions. A taxonomy clarifies how existing methods relate and which settings our theory covers. We validate the theoretical predictions in controlled environments across three settings, where the true symmetry subgroup, perturbation strength, and failure modes can be explicitly controlled.