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This paper introduces the Weak-Entropy PINN (WEPINN) framework, which addresses the challenges of solving hyperbolic conservation laws with discontinuous solutions by enforcing weak formulations of governing equations and incorporating entropy conditions. The method utilizes discrete fast Fourier transform (DFFT) for efficient numerical integration, allowing for accurate resolution of sharp discontinuities and interactions between shock and rarefaction waves. Extensive numerical experiments validate that WEPINN outperforms existing physics-informed neural network approaches by maintaining accuracy without relying on strong prior assumptions or artificial smoothing terms.
Accurately resolving sharp discontinuities in hyperbolic conservation laws is now feasible without the pitfalls of prior assumptions or artificial smoothing.
In recent years, neural networks have significantly advanced numerical solutions of partial differential equations (PDEs). However, solving PDEs with discontinuous solutions, such as hyperbolic conservation laws, remains challenging for neural network-based methods such as physics-informed neural networks (PINNs). Existing methods often rely on strong prior assumptions such as knowledge of discontinuity locations, or they introduce artificial smoothing terms that degrade accuracy. However, accurately solving these conservation laws and predicting the formation and propagation of discontinuities in solutions is crucial in many practical applications, including gas dynamics and traffic flow modeling. In this paper, we introduce a novel Weak-Entropy PINN (WEPINN) framework for hyperbolic conservation laws with discontinuous solutions. The method enforces the governing equations in their weak (integral) formulation and incorporates the entropy condition to select the physically admissible solution, while employing the discrete fast Fourier transform (DFFT) for efficient numerical integration. Our method is tested through extensive numerical experiments on a variety of scalar conservation laws and systems of conservation laws in one and two dimensional spaces. These experiments demonstrate that our method can accurately resolve sharp discontinuities while effectively capturing interactions between multiple shock and rarefaction waves.
