In this paper, we propose a cusp-capturing physics-informed neural network (PINN) to solve discontinuous-coefficient elliptic interface problems whose solution is continuous but has discontinuous first derivatives on the interface. To find such a solution using neural network representation, we introduce a cusp-enforced level set function as an additional feature input to the network to retain the inherent solution properties; that is, capturing the solution cusps (where the derivatives are discontinuous) sharply. In addition, the proposed neural network has the advantage of being mesh-free, so it can easily handle problems in irregular domains. We train the network using the physics-informed framework in which the loss function comprises the residual of the differential equation together with certain interface and boundary conditions. We conduct a series of numerical experiments to demonstrate the effectiveness of the cusp-capturing technique and the accuracy of the present network model. Numerical results show that even using a one-hidden-layer (shallow) network with a moderate number of neurons and sufficient training data points, the present network model can achieve prediction accuracy comparable with traditional methods. Besides, if the solution is discontinuous across the interface, we can simply incorporate an additional supervised learning task for solution jump approximation into the present network without much difficulty.

翻译：本文提出了一种尖点捕捉物理信息神经网络（PINN），用于求解解在界面上连续但一阶导数不连续的间断系数椭圆界面问题。为利用神经网络表示此类解，我们引入了一个尖点强制水平集函数作为网络的附加特征输入，以保留解的固有性质：即锐利地捕捉解尖点（导数不连续处）。此外，所提出的神经网络具有无网格的优点，因此能轻松处理不规则域中的问题。我们采用物理信息框架训练网络，其中损失函数包含微分方程的残差以及特定的界面条件和边界条件。通过一系列数值实验，我们验证了尖点捕捉技术的有效性及所提网络模型的精度。数值结果表明，即使使用单隐层（浅层）网络、适量神经元和足够的训练数据点，所提网络模型也能达到与传统方法相媲美的预测精度。此外，若解在界面上不连续，我们可轻松地将用于解跳跃逼近的额外监督学习任务纳入所提网络，而无需过多困难。