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

2025-08-09

AI Summary

To address modeling errors and accuracy degradation in stochastic partial differential equations (SPDEs) arising from ambiguous physical mechanisms and dynamically changing environments, this paper proposes an adaptive solution framework that synergistically integrates physical priors with data-driven learning. Methodologically, physical constraints are embedded into a score function to construct a training-free ensemble score filter, enabling real-time state updates via recursive Bayesian inference. Furthermore, a likelihood correction mechanism is incorporated into the backward stochastic differential equation, facilitating online adaptive learning from multi-source, sparse, and noisy observations. Compared to conventional approaches, the proposed framework significantly enhances both the accuracy and robustness of SPDE solutions. Numerical experiments across multiple benchmark problems demonstrate its superior stability and computational efficiency.

Technology Category

• Reasoning under Uncertainty: Stochastic Optimization
• Intelligent Robots: State Estimation
• Cognitive Modeling & Cognitive Systems: Adaptive Behavior

Application Category

• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking
• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Systems and Infrastructure for Web, Mobile and WoT: Applied ML and AI for Web-based mobile applications

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.

Problem

• Adaptively learning solutions of stochastic partial differential equations
• Addressing model errors and accuracy in SPDE numerical solutions
• Improving computational efficiency for high-dimensional SPDEs

Innovation

• Score-based diffusion models for SPDEs
• Ensemble score filter for efficiency
• Adaptive learning via Bayesian inference