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

TL;DR

This paper presents a new stochastic operator-learning framework called SON that learns from noisy data to predict solutions and quantify uncertainty in stochastic PDEs, improving modeling of complex physical systems.

Contribution

The paper introduces SON, combining DeepONet and SNNs, with a novel training method for effective uncertainty quantification in SPDEs from noisy data.

Findings

SON accurately captures solution structure in benchmark SPDEs.

The method robustly quantifies predictive uncertainty.

Numerical experiments demonstrate improved performance over existing methods.

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.