  We propose a novel framework for discovering Stochastic Partial Differential
Equations (SPDEs) from data. The proposed approach combines the concepts of
stochastic calculus, variational Bayes theory, and sparse learning. We propose
the extended Kramers-Moyal expansion to express the drift and diffusion terms
of an SPDE in terms of state responses and use Spike-and-Slab priors with
sparse learning techniques to efficiently and accurately discover the
underlying SPDEs. The proposed approach has been applied to three canonical
SPDEs, (a) stochastic heat equation, (b) stochastic Allen-Cahn equation, and
(c) stochastic Nagumo equation. Our results demonstrate that the proposed
approach can accurately identify the underlying SPDEs with limited data. This
is the first attempt at discovering SPDEs from data, and it has significant
implications for various scientific applications, such as climate modeling,
financial forecasting, and chemical kinetics.
