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

TL;DR

This paper introduces iUzawa-Net, a deep learning framework that unrolls an inexact Uzawa method to enable real-time solutions for nonsmooth linear PDE optimal control problems, combining optimization theory and neural networks.

Contribution

The paper presents the first neural network-based solver for nonsmooth linear PDE optimal control problems, unrolling an inexact Uzawa method with proven approximation and optimality properties.

Findings

Achieves real-time solutions for nonsmooth PDE control problems.

Demonstrates promising numerical efficiency on elliptic and parabolic cases.

Provides a versatile framework combining optimization algorithms and deep learning.

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.