A coordinate transformation-based physics-informed neural networks for hyperbolic conservation laws

Hyperbolic conservation laws play a critical role in various fields, including aerodynamics, physics, and oceanography. However, traditional physics-informed neural networks (PINNs), despite their remarkable capabilities in solving partial differential equations (PDEs), often struggle to accurately resolve these problems. To address this challenge, a coordinate transformation-based PINN (CT-PINN) algorithm for hyperbolic conservation laws is proposed, which uses coordinate transformations along characteristic curves to prevent the generation and propagation of discontinuities. The coordinate transformation transforms subdomains divided along characteristic curves into regular domains governed by the corresponding transformed PDEs. The CT-PINN framework simultaneously learns the characteristic curves and the transformed solutions by optimizing a loss function that integrates both the transformed PDEs and the characteristic equations. Due to the equivalence between solutions in the transformed and original domains, predictions in arbitrary coordinates can be obtained without the need for interpolation. Moreover, different PINN architectures can be applied for each subdomain, with hyperparameters flexibly adjusted to enhance accuracy. The proposed method has been evaluated on a range of hyperbolic conservation laws, including the convection equation, the Burgers equation, the shallow water wave equation, the traffic flow equation and the Euler equation. The results demonstrate that CT-PINN can accurately solve the characteristic equation and PDEs, and effectively capture shock waves without transition points, outperforming traditional numerical approaches.

Journal of Computational Physics