用于求解双曲守恒律的高效弱熵物理信息神经网络（WEPINN）

中文摘要 AI 辅助

近年来，神经网络极大地推动了偏微分方程（PDE）数值解法的发展，但对于解存在不连续性的PDE，如双曲守恒律，基于神经网络的方法（如物理信息神经网络（PINNs））仍存在挑战。现有方法往往依赖强先验假设（如已知不连续位置），或引入人工平滑项降低精度。然而，在气体动力学、交通流建模等诸多实际应用中，准确求解此类守恒律并预测解中不连续的形成与传播至关重要。本文提出一种新型弱熵物理信息神经网络（WEPINN）框架，用于求解具有不连续解的双曲守恒律。该方法以弱（积分）形式约束控制方程，融入熵条件以选取符合物理规律的解，同时采用离散快速傅里叶变换（DFFT）实现高效数值积分。通过对一维、二维各类标量守恒律及守恒律系统开展大量数值实验，结果表明，该方法可精确分辨尖锐不连续，同时有效捕捉多激波与稀疏波之间的相互作用。

英文摘要

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.

发表机构

• Columbia University（哥伦比亚大学）
• The Chinese University of Hong Kong（香港中文大学）

机构由 AI 辅助整理，请以论文原文为准。

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27 pages, 7 figures