Learning to Control: The iUzawa-Net for Nonsmooth Optimal Control of Linear PDEs
2026-02-12 • Machine Learning
Machine Learning
AI summaryⓘ
The authors introduce iUzawa-Net, a new deep learning method designed to quickly solve tough math problems called nonsmooth optimal control problems involving partial differential equations (PDEs). Their approach combines traditional optimization algorithms with neural networks to replace complex solver steps, making it faster. They show that iUzawa-Net can approximate solutions well and demonstrate its effectiveness on test problems. This work blends known math techniques with machine learning for better problem-solving in this area.
optimal controlpartial differential equationsnonsmooth optimizationdeep neural networksUzawa methodsaddle point problempreconditionerselliptic PDEparabolic PDEuniversal approximation
Authors
Yongcun Song, Xiaoming Yuan, Hangrui Yue, Tianyou Zeng
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.