用神经网络直接求解障碍问题的最优控制，省去反复求解子问题。
A Single-Loop Bilevel Deep Learning Method for Optimal Control of Obstacle Problems
- 用嵌入约束的神经网络同时逼近状态与控制，保持双层结构。
- 提出单循环随机算法，避免嵌套优化，计算成本显著降低。
- 适用于高维复杂区域，对不规则障碍也有效，适合工程优化场景。
障碍问题的最优控制在众多应用中出现，但因其非光滑性、非线性和双层结构而计算困难。传统数值方法依赖网格离散化，通常需重复求解代价高昂的子问题。本文提出一种无网格的单循环双层深度学习方法，可扩展至高维和复杂域，避免重复求解离散化子问题。该方法采用约束嵌入神经网络近似状态与控制，并保留双层结构。为高效训练神经网络，提出单循环随机一阶双层算法（S2-FOBA），消除嵌套优化，且不依赖下层解的唯一性假设。在温和假设下分析了S2-FOBA的收敛性。在典型基准测试中，包括分布控制与障碍控制问题，针对规则与不规则障碍在复杂域上的实验表明，该方法在保持良好精度的同时，相比经典数值方法显著降低了计算成本。
原文摘要 · Abstract (English)
Optimal control of obstacle problems arises in a wide range of applications and is computationally challenging due to its nonsmoothness, nonlinearity, and bilevel structure. Classical numerical approaches rely on mesh-based discretization and typically require solving a sequence of costly subproblems. In this work, we propose a single-loop bilevel deep learning method, which is mesh-free, scalable to high-dimensional and complex domains, and avoids repeated solution of discretized subproblems. The method employs constraint-embedding neural networks to approximate the state and control and preserves the bilevel structure. To train the neural networks efficiently, we propose a Single-Loop Stochastic First-Order Bilevel Algorithm (S2-FOBA), which eliminates nested optimization and does not rely on restrictive lower-level uniqueness assumptions. We analyze the convergence behavior of S2-FOBA under mild assumptions. Numerical experiments on benchmark examples, including distributed and obstacle control problems with regular and irregular obstacles on complex domains, demonstrate that the proposed method achieves satisfactory accuracy while reducing computational cost compared to classical numerical methods.
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