Discovering stochastic partial differential equations from limited data using variational Bayes inference
This work addresses the challenge of modeling complex systems like climate and finance by enabling SPDE discovery from data, representing a first attempt in this area.
The authors tackled the problem of discovering Stochastic Partial Differential Equations (SPDEs) from limited data by proposing a novel framework that combines stochastic calculus, variational Bayes theory, and sparse learning, and demonstrated accurate identification of underlying SPDEs in three canonical examples.
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.