Least-Squares ReLU Neural Network (LSNN) Method For Scalar Nonlinear Hyperbolic Conservation Law
This work addresses numerical challenges in hyperbolic PDEs for computational science, presenting an incremental improvement over prior LSNN methods.
The paper tackles solving scalar nonlinear hyperbolic conservation laws with discontinuous solutions by introducing the least-squares ReLU neural network (LSNN) method, which outperforms mesh-based methods in terms of degrees of freedom and avoids Gibbs phenomena at discontinuities.
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 the free breaking lines of the ReLU neural network. Moreover, the method does not exhibit the common Gibbs phenomena along the discontinuous interface.