Geometric regularization of autoencoders via observed stochastic dynamics

Stochastic dynamical systems with slow or metastable behavior evolve, on long time scales, on an unknown low-dimensional manifold in high-dimensional ambient space. Building a reduced simulator from short-burst ambient ensembles is a long-standing problem: local-chart methods like ATLAS suffer from exponential landmark scaling and per-step reprojection, while autoencoder alternatives leave tangent-bundle geometry poorly constrained, and the errors propagate into the learned drift and diffusion. We observe that the ambient covariance~Lambda already encodes coordinate-invariant tangent-space information, its range spanning the tangent bundle. Using this, we construct a tangent-bundle penalty and an inverse-consistency penalty for a three-stage pipeline (chart learning, latent drift, latent diffusion) that learns a single nonlinear chart and the latent SDE. The penalties induce a function-space metric, the ρ-metric, strictly weaker than the Sobolev H1 norm yet achieving the same chart-quality generalization rate up to logarithmic factors. For the drift, we derive an encoder-pullback target via Ito's formula on the learned encoder and prove a bias decomposition showing the standard decoder-side formula carries systematic error for any imperfect chart. Under a W2,∞ chart-convergence assumption, chart-level error propagates controllably to weak convergence of the ambient dynamics and to convergence of radial mean first-passage times. Experiments on four surfaces embedded in up to 201 ambient dimensions reduce radial MFPT error by 50--70% under rotation dynamics and achieve the lowest inter-well MFPT error on most surface--transition pairs under metastable M"uller--Brown Langevin dynamics, while reducing end-to-end ambient coefficient errors by up to an order of magnitude relative to an unregularized autoencoder.