Locally linearized physics-informed neural networks for Riemann problems of hyperbolic conservation laws

In this work, we demonstrate that physics-informed neural networks (PINNs) tend to propagate predicted shock wave information bidirectionally in time, which does not align with the actual evolution direction of solutions to hyperbolic conservation laws. This mismatch results in instability and hinders the reduction of the loss of governing equations, as well as the initial condition loss by the deep neural network. In order to tackle this problem, we simplify the complexity of the problem by constructing equivalent linear transport equations in the region of shock wave generation. The speeds of these linearized waves are governed by the Rankine–Hugoniot relations of conservation laws. This approach is termed the Locally Linearized PINNs method. Specifically, an appropriate shock wave detector is initially designed to identify domains where shock waves occur. Near shock waves, the original nonlinear equations are transformed into their linearized forms, thereby modifying the residual terms of the partial differential equations. Additionally, an equilibrium factor is introduced in fluid compression regions to reduce prediction errors and stabilize the training of deep neural networks. Numerical examples illustrate that Locally Linearized PINNs effectively address the challenge of predicting global solutions in PINNs and significantly improve shock-capturing performance for hyperbolic conservation laws.