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.

翻译：障碍问题的最优控制广泛应用于多个领域，但由于其非光滑性、非线性及双层结构，计算上具有挑战性。经典数值方法依赖于基于网格的离散化，通常需要求解一系列计算代价高昂的子问题。本文提出一种单循环双层深度学习方法，该方法无需网格划分，可扩展至高维复杂区域，且避免了重复求解离散化子问题。该方法采用约束嵌入神经网络来逼近状态和控制变量，并保持了双层结构。为高效训练神经网络，我们提出一种单循环随机一阶双层算法（S2-FOBA），该算法消除了嵌套优化过程，且不依赖于限制性的下层唯一性假设。我们在温和假设下分析了S2-FOBA的收敛行为。在基准算例上的数值实验（包括复杂区域上含规则与非规则障碍的分布式控制及障碍控制问题）表明，与经典数值方法相比，所提方法在保证满意精度的同时显著降低了计算成本。