Deep Uzawa for PDE constrained optimisation

In this work, we present a numerical solver for optimal control problems constrained by linear and semi-linear second-order elliptic PDEs. The approach is based on recasting the problem and includes an extension of Uzawa's algorithm to build approximating sequences for these constrained optimal control problems. We prove strong convergence of the iterative scheme in their respective norms, and this convergence is generalised to a class of restricted function spaces. We showcase the algorithm by demonstrating its use numerically with neural network methods that we coin Deep Uzawa Algorithms and show they perform favourably compared with some existing Deep Neural Network approaches.