In [4], we introduced the least-squares ReLU neural network (LSNN) method for solving the linear advection-reaction problem with discontinuous solution and showed that the number of degrees of freedom for the LSNN method is significantly less than that of traditional mesh-based methods. The LSNN method is a discretization of an equivalent least-squares (LS) formulation in the class of neural network functions with the ReLU activation function; and evaluation of the LS functional is done by using numerical integration and proper numerical differentiation. By developing a novel finite volume approximation (FVA) to the divergence operator, this paper studies the LSNN method for scalar nonlinear hyperbolic conservation laws. The FVA introduced in this paper is tailored to the LSNN method and is more accurate than traditional, well-studied FV schemes used in mesh-based numerical methods. Numerical results of some benchmark test problems with both convex and non-convex fluxes show that the finite volume LSNN (FV-LSNN) method is capable of computing the physical solution for problems with rarefaction waves and capturing the shock of the underlying problem automatically through the free hyper-planes of the ReLU neural network. Moreover, the method does not exhibit the common Gibbs phenomena along the discontinuous interface.
翻译:在 [4] 中,我们采用了最小平方 ReLU 神经网络(LSNN) 方法,用不连续的解决方案解决线性对冲反应问题,并表明LSNN方法的自由度大大低于传统的网基方法。LSNN方法是神经网络功能类中等同的最小方(LS)配方的分解,与RELU激活功能相配合;对LS功能的评估是通过数字整合和适当的数字变异来进行的。通过向差异操作员开发新的定量量近似(FVA),本文研究了LSNN用于SNNN方法的自由度比传统的网基方法要低得多。 LSNN(F-LSNN) 方法的定量量量量性测试结果显示,通过SNNN(FV-LSNNN) 和不相容的数值变异性通方法,可以自动地计算超线性平面的硬面方法,而不能自动计算超度平面的平面系统问题。