We show that the physics-informed neural networks (PINNs), in combination with some recently developed discontinuity capturing neural networks, can be applied to solve optimal control problems subject to partial differential equations (PDEs) with interfaces and some control constraints. The resulting algorithm is mesh-free and scalable to different PDEs, and it ensures the control constraints rigorously. Since the boundary and interface conditions, as well as the PDEs, are all treated as soft constraints by lumping them into a weighted loss function, it is necessary to learn them simultaneously and there is no guarantee that the boundary and interface conditions can be satisfied exactly. This immediately causes difficulties in tuning the weights in the corresponding loss function and training the neural networks. To tackle these difficulties and guarantee the numerical accuracy, we propose to impose the boundary and interface conditions as hard constraints in PINNs by developing a novel neural network architecture. The resulting hard-constraint PINNs approach guarantees that both the boundary and interface conditions can be satisfied exactly or with a high degree of accuracy, and they are decoupled from the learning of the PDEs. Its efficiency is promisingly validated by some elliptic and parabolic interface optimal control problems.

翻译：我们证明，物理信息神经网络（PINNs）结合近期发展的一些间断捕捉神经网络，可用于求解具有界面及控制约束的偏微分方程（PDE）最优控制问题。所提出的算法无需网格划分，可扩展至不同类型的PDE，并能严格保证控制约束。由于边界条件、界面条件以及PDE本身均作为软约束被合并至加权损失函数中，必须同时学习这些条件，且无法保证边界和界面条件被精确满足。这直接导致对应损失函数中权重调整与神经网络训练面临困难。为解决这些难题并保证数值精度，我们提出通过构建一种新颖的神经网络架构，将边界与界面条件作为硬约束嵌入PINNs中。由此产生的硬约束PINNs方法能够确保边界和界面条件被精确或高精度满足，且这些条件与PDE的学习过程解耦。通过若干椭圆型和抛物型界面最优控制问题的算例，该方法的有效性得到了有力验证。