The paper presents a new framework called the Stochastic Operator Network (SON) for uncertainty quantification of solution operators in stochastic partial differential equations (SPDEs). SON combines the Deep Operator Network architecture with Stochastic Neural Networks to learn directly from noisy data, producing both a mean solution field and a measure of predictive uncertainty. The training uses a Hamiltonian-type loss optimized via the Stochastic Maximum Principle, and numerical experiments on benchmark SPDEs show the method’s accuracy and robustness.
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