用随机神经网络学习不确定性的微分方程解，直接从噪声数据中预测结果和不确定性。
Diffusion-Based Stochastic Operator Networks for Uncertainty Quantification in Stochastic Partial Differential Equations
- 结合DeepONet与随机神经网络，构建可概率预测的随机算子网络
- 在多源不确定性下，准确捕捉解结构并量化预测置信度
- 适合需要可靠不确定性估计的物理建模与科学计算场景
我们提出一种新框架，用于量化随机偏微分方程（SPDE）解算子的不确定性。尽管SPDE在不确定性下的复杂物理系统建模中至关重要，但其实际应用通常需预先设定模型不确定性的大小与结构，而这些往往未知且难以从噪声观测中推断。为此，我们开发了一种直接从噪声数据中学习的随机算子学习框架，输出均值解场及不确定性量化结果。所提方法——随机算子网络（SON），通过融合深度算子网络（DeepONet）与随机神经网络（SNNs）的结构，建模随机性并实现概率预测。训练过程通过最小化哈密顿型损失，并利用随机最大原理优化目标。在多个基准SPDE及多源不确定性下的数值实验表明，该方法在捕捉解结构和量化预测不确定性方面具有高精度与鲁棒性。
原文摘要 · Abstract (English)
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.
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