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

Abstract

We develop a novel framework for uncertainty quantification in operator learning, namely the stochastic operator network (SON). SON combines the stochastic optimal control concepts of the stochastic neural network (SNN) with the deep operator network. 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 the stochastic maximum principle in the stochastic gradient descent (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.