Capturing Shock Waves by Relaxation Neural Networks
This addresses a specific bottleneck in computational physics for researchers dealing with hyperbolic systems, offering an incremental improvement over PINN.
The paper tackles the problem of solving nonlinear hyperbolic systems with shock waves using physics-informed neural networks (PINN), which often fail due to optimization issues from discontinuities. The proposed Relaxation Neural Networks (RelaxNN) framework alleviates these conflicts and demonstrates remarkable results in numerical simulations.
In this paper, we put forward a neural network framework to solve the nonlinear hyperbolic systems. This framework, named relaxation neural networks(RelaxNN), is a simple and scalable extension of physics-informed neural networks(PINN). It is shown later that a typical PINN framework struggles to handle shock waves that arise in hyperbolic systems' solutions. This ultimately results in the failure of optimization that is based on gradient descent in the training process. Relaxation systems provide a smooth asymptotic to the discontinuity solution, under the expectation that macroscopic problems can be solved from a microscopic perspective. Based on relaxation systems, the RelaxNN framework alleviates the conflict of losses in the training process of the PINN framework. In addition to the remarkable results demonstrated in numerical simulations, most of the acceleration techniques and improvement strategies aimed at the standard PINN framework can also be applied to the RelaxNN framework.