Diffusion-Based Stochastic Operator Networks for Uncertainty Quantification in Stochastic Partial Differential Equations

2026-05-16

AI Summary

This work addresses the challenge of characterizing model uncertainty and handling noisy observational data in stochastic partial differential equations (SPDEs) by proposing a novel stochastic operator learning framework. The method integrates deep operator networks (DeepONets) with stochastic neural networks to construct an end-to-end trainable Stochastic Operator Network (SON). Leveraging a Hamiltonian-inspired loss function and optimization grounded in the stochastic maximum principle, the framework simultaneously predicts the mean solution field and quantifies predictive uncertainty. Evaluated on benchmark SPDE problems involving multiple sources of uncertainty, the approach not only accurately reconstructs the solution structure but also robustly estimates uncertainty, thereby achieving joint modeling of both the SPDE solution and its associated uncertainties.

Abstract

We introduce a novel framework for uncertainty quantification of solution operators associated with stochastic partial differential equations (SPDEs). Although SPDEs play a central role in modeling complex physical systems under uncertainty, their practical use typically requires specifying the magnitude and structure of model uncertainties that are often unknown and difficult to infer from noisy measurements. To address this challenge, we develop a stochastic operator-learning framework that learns directly from noisy data and outputs both a mean solution field and a quantification of uncertainty. The proposed method, namely the Stochastic Operator Network (SON), is constructed by combining the structure of the Deep Operator Network (DeepONet) with Stochastic Neural Networks (SNNs) to model stochasticity and enable probabilistic prediction. The training procedure is carried out by minimizing a Hamiltonian-type loss and optimizing the resulting objective using the Stochastic Maximum Principle. Numerical experiments on benchmark SPDEs under multiple uncertainty sources demonstrate the accuracy and robustness of the proposed method in capturing solution structure and quantifying predictive uncertainty.

Problem

• stochastic partial differential equations
• uncertainty quantification
• solution operators
• noisy data
• model uncertainty

Innovation

• Stochastic Operator Network
• Uncertainty Quantification
• Stochastic Partial Differential Equations
• Deep Operator Network
• Stochastic Neural Networks