Discontinuity Computing Using Physics-Informed Neural Networks

Simulating discontinuities has been a long-standing challenge, especially when dealing with shock waves characterized by strong nonlinear features. Despite their promise, the recently developed physics-informed neural networks (PINNs) have not yet fully demonstrated their effectiveness in handling discontinuities when compared to traditional shock-capturing methods. In this study, we reveal a paradoxical phenomenon during the training of PINNs when computing problems with strong nonlinear discontinuities. To address this issue and enhance the PINNs’ ability to capture shocks, we propose PINNs-WE (Physics-Informed Neural Networks with Equation Weight) method by introducing three novel strategies. Firstly, we attenuate the neural network’s expression locally at ‘transition points’ within the shock waves by introducing a physics-dependent weight into the governing equations. Consequently, the neural network will concentrate on training the smoother parts of the solutions. As a result, due to the compressible property, sharp discontinuities emerge, with transition points being compressed into well-trained smooth regions akin to passive particles. Secondly, we also introduce the Rankine–Hugoniot (RH) relation, which is equivalent to the weak form of the conservation laws near the discontinuity, in order to improve the shock-capturing preformance. Lastly, we construct a global physical conservation constraint to enhance the conservation properties of PINNs which is key to resolve the right position of the discontinuity. To illustrate the impact of our novel approach, we investigate the behavior of the one-dimensional Burgers’ equation, as well as the one- and two-dimensional Euler equations. In our numerical experiments, we compare our proposed PINNs-WE method with a traditional high-order weighted essential non-oscillatory (WENO) approach. The results of our study highlight the significant enhancement in discontinuity computing by the PINNs-WE method when compared to traditional PINNs.