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

2026-02-12

AI Summary

This work addresses the challenge of efficiently solving nonsmooth linear partial differential equation (PDE)-constrained optimal control problems in real time by proposing iUzawa-Net, a novel framework that unfolds the inexact Uzawa algorithm into a deep neural network. For the first time, this approach integrates model-based optimization with data-driven learning, replacing conventional preconditioners and PDE solvers with learnable neural components. The resulting solver combines universal approximation capability with asymptotic ε-optimality, enabling high-accuracy real-time control. Numerical experiments on nonsmooth elliptic and parabolic optimal control problems demonstrate that iUzawa-Net substantially improves computational efficiency while maintaining solution accuracy, making it well-suited for real-time applications.

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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.

Problem

• nonsmooth optimal control
• linear PDEs
• real-time solution
• saddle point problems
• PDE-constrained optimization

Innovation

• iUzawa-Net
• nonsmooth optimal control
• PDE-constrained optimization
• algorithm unrolling
• deep learning