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

• URL: http://arxiv.org/abs/2508.06834v1
• Date: Sat, 09 Aug 2025 05:24:21 GMT
• Title: A Score-based Diffusion Model Approach for Adaptive Learning of Stochastic Partial Differential Equation Solutions
• Authors: Toan Huynh, Ruth Lopez Fajardo, Guannan Zhang, Lili Ju, Feng Bao,
• Abstract summary: We propose a framework for adaptively learning the time-evolving solutions of partial differential equations (SPDEs)<n>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 differential equation.<n>This enables adaptive learning through iterative refinement of the solution as new data becomes available.
• Score: 12.568066880515977
• License: http://creativecommons.org/licenses/by/4.0/
• 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.

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