Problem
Research questions and friction points this paper is trying to address.
hyperbolic conservation laws
discontinuous solutions
physics-informed neural networks
entropy condition
weak formulation
Efficient Weak-Entropy PINN for Solving Hyperbolic Conservation Laws
🤖 AI Summary
This work addresses the challenge of solving hyperbolic conservation laws with discontinuous solutions, where conventional physics-informed neural networks (PINNs) often rely on prior knowledge of discontinuity locations or introduce artificial dissipation that compromises accuracy. The authors propose the Weak Entropy Physics-Informed Neural Network (WEPINN), which for the first time integrates the weak-form PDE constraints and entropy conditions directly into the PINN framework. By incorporating discrete fast Fourier transforms for efficient numerical integration, WEPINN eliminates the need for prior information about shock locations. The method accurately captures the formation and complex interactions of shocks and rarefaction waves in both two-dimensional scalar and system conservation laws, resolving sharp discontinuities while maintaining high solution accuracy.
📝 Abstract
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.
Innovation
Methods, ideas, or system contributions that make the work stand out.
Weak-Entropy PINN
hyperbolic conservation laws
discontinuous solutions
entropy condition
discrete fast Fourier transform
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