Stoch-IDENT: New Method and Mathematical Analysis for Identifying SPDEs from Data

Abstract: In this paper, we propose Stoch-IDENT, a novel method for identifying Stochastic Partial Differential Equations (SPDEs) from observational data. Our method can handle linear and nonlinear high-order SPDEs driven by time-dependent Wiener processes with both additive or multiplicative structures. Theoretically, we establish a rigorous connection between the spectral properties of the solution's mean and covariance and the identifiability of the underlying SPDEs. Our analysis covers key classes of equations, including linear SPDEs with constant coefficients, as well as parabolic and hyperbolic types, generalizing the theory of identification of deterministic PDEs. Algorithmically, the drift term is identified using a sample mean generalization of Robust-IDENT (He et al., 2023). For the diffusion term, we develop a new greedy algorithm, Quadratic Subspace Pursuit (QSP), which can address general sparse regression problems with quadratic measurements. We validate Stoch-IDENT extensively on various SPDEs, demonstrating its effectiveness through quantitative and qualitative evaluations.