Neural Networks with Local Converging Inputs (NNLCI) for Solving Conservation Laws, Part I: 1D Problems (Communications in Computational Physics, 2023)
A novel neural network method is developed for solving systems of conservation laws whose solutions may contain abrupt changes of state, including shockwaves and contact discontinuities. In conventional approaches, a low-cost solutionpatch is usually used as the input to a neural network for predicting the high-fidelitysolution patch. With that technique, however, there is no way to distinguish a smeareddiscontinuity from a smooth solution with large gradient in the input, and the two almost identical inputs correspond to two fundamentally different high-fidelity solutionpatches in training and predicting. To circumvent this difficulty, we use local patchesof two low-cost numerical solutions of the conservation laws in a converging sequenceas the input to a neural network. The neural network then makes a correct predictionby identifying whether the solution contains discontinuities or just smooth variationswith large gradients, because the former becomes increasingly steep in a convergingsequence in the input, and the latter does not. The inputs can be computed from low-cost numerical schemes with coarse resolution, in a local domain of dependence of aspace-time location where the prediction is to be made. Despite smeared input solutions, the output provides sharp approximations of solutions containing shock wavesand contact discontinuities. The method works effectively not only for regions withdiscontinuities, but also for smooth regions of the solution. It is efficient to implement,once trained, and has broader applications for different types of differential equations.