A Score-based Diffusion Model Approach for Adaptive Learning of Stochastic Partial Differential Equation Solutions
This work addresses model errors and reduced accuracy in SPDE solutions for complex physical systems, offering an adaptive learning approach that is incremental in improving computational efficiency for high-dimensional settings.
The paper tackles the problem of learning time-evolving solutions for stochastic partial differential equations (SPDEs) under uncertainty by proposing a novel framework using score-based diffusion models with recursive Bayesian inference, achieving accurate and robust results in numerical experiments on benchmark SPDEs with sparse and noisy observations.
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.