In this paper, a new Discontinuity Capturing Shallow Neural Network (DCSNN) for approximating $d$-dimensional piecewise continuous functions and for solving elliptic interface problems is developed. There are three novel features in the present network; namely, (i) jump discontinuities are accurately captured, (ii) it is completely shallow, comprising only one hidden layer, (iii) it is completely mesh-free for solving partial differential equations. The crucial idea here is that a $d$-dimensional piecewise continuous function can be extended to a continuous function defined in $(d+1)$-dimensional space, where the augmented coordinate variable labels the pieces of each sub-domain. We then construct a shallow neural network to express this new function. Since only one hidden layer is employed, the number of training parameters (weights and biases) scales linearly with the dimension and the neurons used in the hidden layer. For solving elliptic interface problems, the network is trained by minimizing the mean square error loss that consists of the residual of the governing equation, boundary condition, and the interface jump conditions. We perform a series of numerical tests to demonstrate the accuracy of the present network. Our DCSNN model is efficient due to only a moderate number of parameters needed to be trained (a few hundred parameters used throughout all numerical examples), and the results indicate good accuracy. Compared with the results obtained by the traditional grid-based immersed interface method (IIM), which is designed particularly for elliptic interface problems, our network model shows a better accuracy than IIM. We conclude by solving a six-dimensional problem to demonstrate the capability of the present network for high-dimensional applications.
翻译:在本文中, 一个新的 Discondition Capture Shallow Neal 网络( DCSNNN ), 用于 $dd$+1美元 的立体连续功能, 并用于解决椭圆界面问题 。 在目前的网络中, 有三个新特点 : (一) 精确地捕捉跳不连续现象, (二) 完全浅色, 仅包含一个隐藏层, 仅包含一个隐藏层, (三) 完全无网外观解决部分差异方程式。 这里的关键思想是, 一个以美元为维平面平面的常规界面持续功能可以扩展到一个在$( d+1) 维空间中定义的连续函数, 在那里, 增强的坐标坐标坐标可给每个子目录的片段贴标签。 我们随后建造了一个浅色的神经网络网络来表达这一新功能。 由于只使用一个隐藏层, 线性参数( 重量和偏差) 和隐藏层中使用的神经等。 为了解决极型界面的界面问题, 以最小化的平方位错误损失 。 我们使用一个数字测试到整个网络需要一个数字测试, 以100 。