A sparse Bayesian framework for discovering interpretable nonlinear stochastic dynamical systems with Gaussian white noise

Abstract

Extracting governing physics from data is a key challenge in many areas of science and technology. The existing techniques for equation discovery are mostly applicable to deterministic systems and require both input and state measurements. We here propose a novel data-driven framework for discovering nonlinear stochastic dynamical systems with Gaussian white noise. The proposed framework blends concepts of stochastic calculus, sparse learning algorithms, and Bayesian statistics to learn the governing physics from data. In particular, we combine sparsity–promoting spike and slab prior, Bayes law, and the Kramers–Moyal formula to identify stochastic differential equations from data. The proposed framework is highly efficient and works with sparse, noisy, and incomplete output measurements. The efficacy and robustness of the proposed approach are illustrated in several numerical examples involving both complete and partial state measurements. The results obtained indicate the potential of the proposed approach in discovering nonlinear stochastic dynamical systems subjected to Gaussian white noise excitation.