Learning to Control: The iUzawa-Net for Nonsmooth Optimal Control of Linear PDEs

Abstract

We propose an optimization-informed deep neural network approach, named iUzawa-Net, aiming for the first solver that enables real-time solutions for a class of nonsmooth optimal control problems of linear partial differential equations (PDEs). The iUzawa-Net unrolls an inexact Uzawa method for saddle point problems, replacing classical preconditioners and PDE solvers with specifically designed learnable neural networks. We prove universal approximation properties and establish the asymptotic $\varepsilon$-optimality for the iUzawa-Net, and validate its promising numerical efficiency through nonsmooth elliptic and parabolic optimal control problems. Our techniques offer a versatile framework for designing and analyzing various optimization-informed deep learning approaches to optimal control and other PDE-constrained optimization problems. The proposed learning-to-control approach synergizes model-based optimization algorithms and data-driven deep learning techniques, inheriting the merits of both methodologies.