An Operator Learning Approach to Nonsmooth Optimal Control of Nonlinear PDEs

Abstract

Optimal control problems with nonsmooth objectives and nonlinear partial differential equation (PDE) constraints are challenging, mainly because of the underlying nonsmooth and nonconvex structures and the demanding computational cost for solving multiple high-dimensional and ill-conditioned systems after mesh-based discretization. To mitigate these challenges numerically, we propose an operator learning approach in combination with an effective primal-dual optimization idea which can decouple the treatment of the control and state variables so that each of the resulting iterations only requires solving two PDEs. Our main purpose is to construct neural surrogate models for the involved PDEs by operator learning, allowing the solution of a PDE to be obtained with only a forward pass of the neural network. The resulting algorithmic framework offers a hybrid approach that combines the efficiency and generalization of operator learning with the model-based nature and structure-friendly efficiency of primal-dual-based algorithms. The primal-dual-based operator learning approach offers numerical methods that are mesh-free, easy to implement, and adaptable to various optimal control problems with nonlinear PDEs. It is notable that the neural surrogate models can be reused across iterations and parameter settings, hence retraining of neural networks can be avoided and computational cost can be substantially alleviated. We affirmatively validate the efficiency of the primal-dual-based operator learning approach across a range of typical optimal control problems with nonlinear PDEs.