A Score-based Diffusion Model Approach for Adaptive Learning of Stochastic Partial Differential Equation Solutions

In this paper, we introduce a score-based diffusion model appr oach for adaptively learning the time-evolving solutions of stochastic partial differential equat ions (SPDEs) through recursive Bayesian inference. Partial differential equations (PDEs) are fundamental tools for modeling the dynamic behavior of complex physical systems. While they have been widely suc cessful in scientific and engineering applications, many practical scenarios involve inherent unc ertainties due to limited physical knowledge and environmental variability. For example, in climate and meteorological modeling, uncertainties in initial conditions, boundary data, and subgrid-scale ph ysical processes can significantly affect the accuracy of predictions governed by PDEs such as the Navier-Stokes or advection-diffusion equations. Similarly, in porous media flow problems, spatial het erogeneity and limited characterization of subsurface properties -- such as permeability or porosity -- i ntroduce substantial uncertainty into models governed by Darcy's law and related PDEs, making accu rate prediction particularly challenging. To capture these uncertainty effects and support rel iable predictive analysis, it is essential to incorporate SPDEs into mathematical modeling framework . The numerical solution of SPDEs has thus become a central focus of the uncertainty quantifica tion (UQ) community, where significant efforts have been dedicated to developing efficient solvers tha t can accurately characterize and propagate uncertainty in high-dimensional, nonlinear dynamica l systems (see, e.g., [1, 2, 13, 21, 36, 42, 53] and the reference therein). Despite advances in SPDE solvers capable of quantifying unc ertainty, significant challenges remain.