Least-Squares ReLU Neural Network (LSNN) Method For Scalar Nonlinear Hyperbolic Conservation Law

Least-Squares ReLU Neural Network (LSNN) Method For Scalar Nonlinear Hyperbolic Conservation Law
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DOI：
10.1016/j.apnum.2022.01.002
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发表时间：
2021-05
期刊：
ArXiv
影响因子：
--
通讯作者：
Z. Cai;Jingshuang Chen;Min Liu
Z. Cai;Jingshuang Chen;Min Liu
中科院分区：
其他
文献类型：
--
作者：
Z. Cai;Jingshuang Chen;Min Liu

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在[7]中，我们引入了最小二乘ReLU神经网络（LSNN）方法来解决具有不连续解的线性平流反应问题，并表明该方法在自由度数方面优于基于网格的数值方法。本文研究了标量非线性双曲守恒定律的LSNN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 (LS) 公式进行离散化。 LS 泛函的评估是通过使用数值积分和保守的有限体积方案来完成的。一些测试问题的数值结果表明，该方法能够通过ReLU神经网络的自由断线自动逼近底层问题的不连续界面。此外，该方法不会沿着不连续界面表现出常见的吉布斯现象。
In [7], we introduced the least-squares ReLU neural network (LSNN) method for solving the linear advection-reaction problem with discontinuous solution and showed that the method outperforms mesh-based numerical methods in terms of the number of degrees of freedom. This paper studies the LSNN method for scalar nonlinear hyperbolic conservation law. The method is a discretization of an equivalent least-squares (LS) formulation in the set of neural network functions with the ReLU activation function. Evaluation of the LS functional is done by using numerical integration and conservative finite volume scheme. Numerical results of some test problems show that the method is capable of approximating the discontinuous interface of the underlying problem automatically through thefreebreaking lines of the ReLU neural network. Moreover, the method does not exhibit the common Gibbs phenomena along the discontinuous interface.