Stochastic Operator Network: A Stochastic Maximum Principle Based Approach to Operator Learning
This addresses uncertainty quantification for operator learning, which is incremental as it builds on existing methods like Stochastic Neural Networks and DeepONet.
The paper tackles uncertainty quantification in operator learning by introducing the Stochastic Operator Network (SON), which combines stochastic optimal control concepts with DeepONet to learn uncertainty through diffusion parameters, demonstrating effectiveness in replicating noisy operators in 2D and 3D.
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.