In this paper, we propose a cusp-capturing physics-informed neural network (PINN) to solve variable-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, 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 a 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 a one-hidden-layer (shallow) network with a moderate number of neurons ($40-60$) and sufficient training data points, the present network model can achieve high prediction accuracy (relative $L^2$ errors in the order of $10^{-5}-10^{-6}$), which outperforms several existing neural network models and traditional grid-based methods in the literature.

翻译：在本文中,我们提出一个快速捕获的物理知情神经网络(PINN),以解决具有持续解决办法、但在界面上有不连续的第一批衍生物的、可变和高效的椭圆界面问题。为了利用神经网络的演示找到这样一个解决方案,我们引入了一个 cupsp加固水平设定功能,作为网络的附加特性输入,以保留内在的解决方案属性,捕捉解决方案(在衍生物不连续的情况下)的快速捕获。此外,拟议神经网络的优势是没有网状的,因此可以很容易地处理非常规域的问题。我们利用物理知情框架对网络进行培训,使损失功能包含差异方程式的剩余部分以及某些界面和边界条件。我们进行了一系列数字实验,以展示刻加固技术的有效性和当前网络模型的准确性。数字结果显示,即使是一层网络(shallow)网络的优势是中等数量的神经元(40-60美元)和足够的培训数据点。目前的网络模型可以实现高精确度的预测值(Restive $-5) 和10-10-10年的网络模型中的若干模型的错误)。