From Initial Data to Boundary Layers: Neural Networks for Nonlinear Hyperbolic Conservation Laws

Abstract--We address the approximation of entropy solutions to initial-boundary value problems for nonlinear strictly hyperbolic conservation laws using neural networks. A general and systematic framework is introduced for the design of efficient and reliable learning algorithms, combining fast convergence during training with accurate predictions. The methodology that relies on solving a certain relaxed related problem is assessed through a series of one-dimensional scalar test cases. These numerical experiments demonstrate the potential of the methodology developed in this paper and its applicability to more complex industrial scenarios. Nonlinear hyperbolic conservation laws play a central role in the mathematical modeling of physical systems where transport and wave propagation phenomena dominate.