Stochastic Operator Network: A Stochastic Maximum Principle Based Approach to Operator Learning

TL;DR

The paper introduces the Stochastic Operator Network (SON), a novel operator learning framework that incorporates stochastic optimal control principles to quantify uncertainty in learned operators.

Contribution

It combines stochastic control with neural operator learning, replacing loss gradients with Hamiltonian gradients for uncertainty quantification.

Findings

Successfully replicates noisy operators in 2D and 3D

Demonstrates effective uncertainty modeling in operator learning

Integrates stochastic maximum principle into neural operator training

Abstract

We develop a novel framework for uncertainty quantification in operator learning, the Stochastic Operator Network (SON). SON combines the stochastic optimal control concepts of the Stochastic Neural Network (SNN) with the DeepONet. By formulating the branch net as an SDE and backpropagating through the adjoint BSDE, we replace the gradient of the loss function with the gradient of the Hamiltonian from Stohastic Maximum Principle in the SGD update. This allows SON to learn the uncertainty present in operators through its diffusion parameters. We then demonstrate the effectiveness of SON when replicating several noisy operators in 2D and 3D.