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

TL;DR

Stoch-IDENT is a new framework for identifying stochastic PDEs from data, capable of handling complex noise structures and high-order equations, with theoretical analysis and a novel greedy algorithm.

Contribution

It introduces a comprehensive method for SPDE identification, including spectral analysis, sparse regression, and the Quadratic Subspace Pursuit algorithm, with proven stability and effectiveness.

Findings

Successfully identifies linear and nonlinear SPDEs from data.

Develops a stable greedy algorithm for non-convex optimization.

Validates the method on various SPDEs with positive results.

Abstract

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.