Least-squares neural network (LSNN) method for scalar nonlinear hyperbolic conservation laws: Discrete divergence operator
Least-squares neural network (LSNN) method for scalar nonlinear hyperbolic conservation laws: Discrete divergence operator
复制标题
用于标量非线性双曲守恒定律的最小二乘神经网络 (LSNN) 方法：离散散度算子
DOI：
10.1016/j.cam.2023.115298
复制
发表时间：
2023
影响因子：
2.4
通讯作者：
Liu, Min
中科院分区：
文献类型：
--
作者：
Cai, Zhiqiang;Chen, Jingshuang;Liu, Min
采用最小二乘神经网络(LSNN)方法求解Cai等人的标量线性和非线性双曲型守恒律(HCL)。(2021、2022)。该方法基于一个等价的最小二乘(LS)公式，并使用RELU神经网络作为逼近函数，这使得它非常适合于逼近具有未知界面位置的不连续函数。在LSNN方法的设计中，微分算子的数值逼近是一个关键因素，而标准的数值或沿坐标方向的自动微分往往会导致基于神经网络的方法失败。为了克服这一挑战，本文将HCL重写为空间和时间的发散形式，并引入了一个新的离散发散算子。理论上，即使在解不连续的情况下，离散发散算子的精度也得到了估计。在数值上，采用新的离散散度算子的LSNN方法对几个既有凸通量又有非凸通量的基准问题进行了测试，对于稀疏波、激波和复合波问题能够计算出正确的物理解。该方法能够捕捉潜在问题的激波而不振荡或涂抹，甚至不需要对熵条件、总变分和/或人工粘性进行任何惩罚。
A least-squares neural network (LSNN) method was introduced for solving scalar linear and nonlinear hyperbolic conservation laws (HCLs) in Cai et al. (2021, 2022). This method is based on an equivalent least-squares (LS) formulation and uses ReLU neural network as approximating functions, making it ideal for approximating discontinuous functions with unknown interface location. In the design of the LSNN method for HCLs, the numerical approximation of differential operators is a critical factor, and standard numerical or automatic differentiation along coordinate directions can often lead to a failed NN-based method. To overcome this challenge, this paper rewrites HCLs in their divergence form of space and time and introduces a new discrete divergence operator. As a result, the proposed LSNN method is free of penalization of artificial viscosity.Theoretically, the accuracy of the discrete divergence operator is estimated even for discontinuous solutions. Numerically, the LSNN method with the new discrete divergence operator was tested for several benchmark problems with both convex and non-convex fluxes, and was able to compute the correct physical solution for problems with rarefaction, shock or compound waves. The method is capable of capturing the shock of the underlying problem without oscillation or smearing, even without any penalization of the entropy condition, total variation, and/or artificial viscosity.
登录
查看更多内容
DOI：
10.1016/j.cma.2013.12.015
发表时间：
2014-04
影响因子：
7.2
作者：
J. Guermond;Murtazo Nazarov
通讯作者：
J. Guermond;Murtazo Nazarov
自适应深度神经网络：函数和偏微分方程的数值逼近
影响因子：
4.1
作者：
Cai, Zhiqiang;Chen, Jingshuang;Liu, Min
通讯作者：
Liu, Min
影响因子：
4.1
作者：
ROE, PL
通讯作者：
ROE, PL
影响因子：
4.1
作者：
Raissi, M.;Perdikaris, P.;Karniadakis, G. E.
通讯作者：
Karniadakis, G. E.
Burgers方程的H(div)相容最小二乘有限元方法的数值守恒性质
影响因子：
3.1
作者：
H. Sterck;T. Manteuffel;S. McCormick;Luke N. Olson
通讯作者：
Luke N. Olson