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.

翻译：我们提出了一种优化信息驱动的深度神经网络方法，命名为iUzawa-Net，旨在成为首个能够实时求解一类线性偏微分方程（PDE）非光滑最优控制问题的求解器。iUzawa-Net展开了一种用于鞍点问题的不精确Uzawa方法，用专门设计的可学习神经网络替代了经典的预条件子和PDE求解器。我们证明了iUzawa-Net的通用逼近性质，并建立了其渐近$\varepsilon$-最优性，同时通过非光滑椭圆和抛物型最优控制问题验证了其优异的数值效率。我们的技术为设计和分析各类用于最优控制及其他PDE约束优化问题的优化信息驱动深度学习方法提供了一个通用框架。所提出的学习控制方法融合了基于模型的优化算法与数据驱动的深度学习技术，兼具两种方法的优点。