Stoch-IDENT: New Method and Mathematical Analysis for Identifying SPDEs from Data
This addresses the challenge of modeling complex spatiotemporal systems with uncertainty for researchers in applied mathematics and computational science, but it is incremental as it builds on existing PDE identification methods.
The authors tackled the problem of identifying stochastic partial differential equations (SPDEs) from observational data, proposing Stoch-IDENT, which handles linear and nonlinear high-order SPDEs with time-dependent noise, and they validated it on various SPDEs with quantitative and qualitative evaluations.
In this paper, we propose Stoch-IDENT, a novel framework 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, accommodating both additive and multiplicative noise structures. To investigate the identifiability of SPDEs from trajectory data, we analyze the spectral properties of the solution's mean and covariance for linear SPDEs with constant coefficients, as well as the dimension of the solution space for parabolic and hyperbolic types, generalizing the identifiability theory for deterministic PDEs. Algorithmically, the drift term is identified via a sample-mean generalization of existing methods for PDE identification. For the diffusion term, we formulate a sparse regression problem with quadratic measurements induced from drift residuals and feature covariances. To address this challenging non-convex and non-smooth optimization, we develop a new greedy algorithm, Quadratic Subspace Pursuit (QSP), and prove that QSP enjoys stable support recovery under certain conditions. We validate Stoch-IDENT on various SPDEs, demonstrating its effectiveness through quantitative and qualitative evaluations.