This paper addresses the problem of approximating the entropy solution to the initial-boundary value problem of a nonlinear strictly hyperbolic conservation law using neural networks. We present a general and systematic framework for designing efficient and reliable learning algorithms that combine fast convergence during training with accurate predictions. We evaluate the methodology for solving specific relaxed related problems using 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.