Gaeta-Lie Neural SDEs: Symmetry-Regularized Learning of Stochastic Dynamics

Learning neural stochastic differential equations (SDEs) from trajectory data is a central task in scientific machine learning, but standard neural SDEs trade flexibility for coefficient fidelity: flexible parameterizations recover coefficients poorly, while structured architectures impose strong coefficient priors. We introduce Gaeta-Lie Neural SDEs, a three-stage pipeline that improves coefficient recovery through discovered symmetries without priors on the drift or diffusion. Building on the Itô SDE symmetry framework of Gaeta and Quintero (1999), we first train a flexible neural SDE surrogate, then discover approximate projectable Lie symmetries of the learned surrogate in a finite basis of canonical symmetry generators, and finally regularize the surrogate by penalizing the residual of the Itô symmetry determining equations. The discovery stage is lightweight, scalable, and theory-guided: the SDE symmetry Lie algebra has known dimensional bounds that constrain the symmetry search space. The resulting regularization is architecture-agnostic and improves existing neural SDE methods when combined with them. Across non-trivial SDEs spanning low and high dimensions and a real-data setting, Gaeta-Lie regularization consistently improves fidelity over multiple baselines, with gains that persist in the data-scarce regime as guaranteed by the symmetry residual's freedom from data sparsity. To our knowledge, this is the first method to use SDE Lie symmetry to improve a learned stochastic surrogate.