Learning Cole-Hopf Coordinates for Stochastic Growth Dynamics: the KPZ class

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 $\mathrm{Var}(h^t)\sim t^{2\beta }$ with $\beta =1/3$ 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.