Learning KPZ Dynamics

Non-linear stochastic processes are notoriously difficult to model, and inferring the dy- namical equations from observations alone can be extremely challenging. To address this, our work develops an entirely data-driven Neural Network framework that learns a trans- form to linearize the system’s dynamics. This is achieved by mapping the observations onto a latent space where the dynamical evolution is of a linear form. The result is a highly interpretable model, as the learnt transformation can be related to known functions and the dynamics to corresponding linear operators. While neural network-based approaches have demonstrated considerable success in modeling deterministic dynamical systems, ex- tending them to the stochastic regime represents a novel research frontier. We develop new ideas and validate them on the Kardar-Parisi-Zhang (KPZ) equation, a paradigmatic model for non-linear stochastic growth. This system is of particular interest because there is a known analytical solution, the Cole-Hopf transformation, which linearizes the dynamics. This allows for a rigorous comparison between our learnt components and the theoretical solution.