Stochastic Latent Twins: Learning Stochastic Solution Operators

Abstract

Scientific prediction under uncertainty requires more than a single future state: it requires the distribution of possible outcomes. We show that this distribution-valued evolution can be learned directly as a stochastic solution operator between arbitrary source and target times. We introduce Stochastic Latent Twins, which represent the physical state through a deterministic encoder-decoder while placing stochasticity entirely in a probability-valued latent evolution map. This separation makes the stochastic transition law independent of the state representation and enables direct probabilistic prediction without recursive time stepping. We establish an end-to-end Wasserstein approximation theory that separates representation and latent-transition errors, derive stability guarantees, and develop a distributional theory of physics conformity with latent-to-physical transfer bounds and conformity by construction. The resulting framework can be learned directly from observed transitions and supports forward prediction, backward conditional evolution, one-shot generation, and physics-conforming stochastic dynamics within the same formulation. Numerical experiments ranging from linear and non-Gaussian SDEs to stochastic Lorenz dynamics, generative modeling, a stochastic Hamiltonian system, and a two-dimensional stochastic Burgers equation verify the theory, expose the distinct roles of state representation and transition-family expressivity, compare favorably with recent stochastic and one-step generative methods, and demonstrate exact physical conservation by construction. More broadly, Stochastic Latent Twins provide a step toward scientific surrogates that represent not only the evolution of a system, but also its uncertainty and physical structure, opening new opportunities across uncertainty quantification, data assimilation, inverse problems, control, and digital twins.