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

TL;DR

This paper introduces a score-based diffusion model framework for adaptively solving stochastic PDEs, integrating physics and observational data for improved accuracy and efficiency in uncertain physical systems.

Contribution

It develops a novel recursive Bayesian approach with an ensemble score filter for real-time, adaptive learning of SPDE solutions, addressing high-dimensional computational challenges.

Findings

Accurate solutions under sparse, noisy data

Robustness demonstrated on benchmark SPDEs

Efficient real-time inference achieved

Abstract

We propose a novel framework for adaptively learning the time-evolving solutions of stochastic partial differential equations (SPDEs) using score-based diffusion models within a recursive Bayesian inference setting. SPDEs play a central role in modeling complex physical systems under uncertainty, but their numerical solutions often suffer from model errors and reduced accuracy due to incomplete physical knowledge and environmental variability. To address these challenges, we encode the governing physics into the score function of a diffusion model using simulation data and incorporate observational information via a likelihood-based correction in a reverse-time stochastic differential equation. This enables adaptive learning through iterative refinement of the solution as new data becomes available. To improve computational efficiency in high-dimensional settings, we introduce the ensemble score filter, a training-free approximation of the score function designed for real-time inference. Numerical experiments on benchmark SPDEs demonstrate the accuracy and robustness of the proposed method under sparse and noisy observations.