Year: 2026
Type: preprint
Abstract: Inferring effective equations from observations of nonlinear stochastic processes is challenging because individual trajectories are noisy realizations, while the relevant physical information is encoded in ensemble statistics. We introduce a data-driven framework that learns a change of variables in which nonlinear stochastic dynamics become linear. The model combines a neural change of representation with a linear stochastic latent evolution and is trained through a short-time moment-matching objective, thereby avoiding trajectory \rev{simulation} during optimization. This construction yields an interpretable generative model: both the learned transformation and the latent evolution operators can be directly inspected. We validate the method on the one-dimensional Kardar--Parisi--Zhang (KPZ) equation, a paradigmatic model of stochastic growth for which the Cole--Hopf transformation maps the nonlinear dynamics to the stochastic heat equation. From simulated height trajectories alone, the model recovers the local exponential Cole--Hopf map and the associated linear stochastic operator. Generated trajectories reproduce the main statistical signatures of the KPZ universality class, including the growth law $\mathrm{Var}(h^{t})\sim t^{2β}$ with $β=1/3$ and the GOE Tracy--Widom height distribution for flat initial conditions. These results show that interpretable stochastic equations can be inferred from data by learning representations in which the dynamics closes in a simple form. Furthermore the development of the physical informed learning permits to get insights on the physics of the dynamical process.
Source: arXiv (Cornell University)
Language: English
Subfield: Statistical and Nonlinear Physics
Field: Physics and Astronomy
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7223306030