用神经网络实时求解线性PDE的非光滑最优控制问题
Learning to Control: The iUzawa-Net for Nonsmooth Optimal Control of Linear PDEs
- 将不精确Uzawa法展开为可学习的神经网络结构
- 在椭圆与抛物型问题上实现ε-最优解，计算效率显著提升
- 适合需快速响应的工业级控制场景，如流体、热传导模拟
我们提出一种基于优化思想的深度神经网络方法iUzawa-Net，旨在首次实现对一类线性偏微分方程（PDE）非光滑最优控制问题的实时求解。该方法将不精确Uzawa法用于鞍点问题，用可学习的神经网络替代传统预条件器和PDE求解器。我们证明了其通用逼近性质，并建立了iUzawa-Net的渐近ε-最优性。通过非光滑椭圆与抛物型最优控制问题验证了其出色的数值效率。所提技术为设计和分析各类优化启发式深度学习方法提供了通用框架，适用于最优控制及其他受PDE约束的优化问题。该学习-控制范式融合了模型驱动优化算法与数据驱动深度学习优势，兼具两者优点。
原文摘要 · Abstract (English)
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.
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