Least-Squares ReLU Neural Network (LSNN) Method For Linear Advection-Reaction Equation
Least-Squares ReLU Neural Network (LSNN) Method For Linear Advection-Reaction Equation
复制标题
DOI：
10.1016/j.jcp.2021.110514
复制
发表时间：
2021-05
期刊：
影响因子：
--
通讯作者：
Z. Cai;Jingshuang Chen;Min Liu
中科院分区：
文献类型：
--
作者：
Z. Cai;Jingshuang Chen;Min Liu
本文研究了求解间断解的线性平流反应问题的最小二乘ReLU神经网络方法。该方法是使用 ReLU 激活函数对神经网络函数集中的等效最小二乘公式进行离散化。该方法能够通过 ReLU 神经网络的自由超平面自动逼近潜在问题的不连续界面，因此在自由度数方面优于基于网格的数值方法。一些基准测试问题的数值结果表明，该方法不仅能用最少的参数逼近解，而且避免了沿不连续界面常见的吉布斯现象。此外，为了很好地逼近 R 2 中界面不是直线的不连续解，三层 ReLU 神经网络是必要且充分的。
This paper studies least-squares ReLU neural network method for solving the linear advection-reaction problem with discontinuous solution. The method is a discretization of an equivalent least-squares formulation in the set of neural network functions with the ReLU activation function. The method is capable of approximating the discontinuous interface of the underlying problem automatically through the free hyper-planes of the ReLU neural network and, hence, outperforms mesh-based numerical methods in terms of the number of degrees of freedom. Numerical results of some benchmark test problems show that the method can not only approximate the solution with the least number of parameters, but also avoid the common Gibbs phenomena along the discontinuous interface. Moreover, a three-layer ReLU neural network is necessary and sufficient in order to well approximate a discontinuous solution with an interface in R 2 that is not a straight line.