基于注意力机制PINNs方法求解圣维南方程

doi:10.16088/j.issn.1001-6600.2024061802

摘要/Abstract

摘要： 针对物理信息神经网络(PINNs)方法在处理时间依赖性问题上的不足,本文提出一种基于注意力机制的物理信息神经网络(PINNsFormer)模拟洪水动态的方法,将PINNsFormer模型应用于求解圣维南方程。PINNsFormer模型能够有效捕捉时空依赖关系,从而提高求解精度和泛化能力。实验结果表明,此方法在模拟洪水传播和捕捉水面剖面细节方面表现出色。在与PINNs以及处理时间特征的神经网络模型FLS和QRes的对比中,PINNsFormer均具有更高的稳定性和精确性。在水平平面和均匀逆坡上的数值试验中,PINNsFormer方法均实现最低的损失值和测试误差,精度达到10^-4量级,准确再现洪水淹没边界的形状。

关键词: 圣维南方程, PINNs, Transformer, 注意力机制

Abstract: A method for simulating flood dynamics using Physics-Informed Neural Networks with Attention Mechanism (PINNsFormer) is proposed to address the shortcomings of Physics-Informed Neural Networks (PINNs) in handling time-dependent problems. The PINNsFormer model is applied to solve the Saint-Venant equations. The model effectively captures spatiotemporal dependencies, thus improving accuracy and generalization. Experimental results show that this method performs excellently in simulating flood propagation and capturing water surface profile details. Compared with PINNs and neural network models FLS and QRes, which handle time features, PINNsFormer demonstrates higher stability and accuracy. Numerical experiments on a horizontal plane and a uniform adverse slope indicate that the PINNsFormer method achieves the lowest loss values and test errors, reaching an accuracy of 10^-4 magnitude, accurately reproducing the shape of flood inundation boundaries.

Key words: Saint-Venant equations, PINNs, transformer, attention mechanism

中图分类号: TP183;O241.8

韩烁, 江林峰, 杨建斌. 基于注意力机制PINNs方法求解圣维南方程[J]. 广西师范大学学报（自然科学版）, 2025, 43(4): 58-68.

HAN Shuo, JIANG Linfeng, YANG Jianbin. Attention-based PINNs Method for Solving Saint-Venant Equations[J]. Journal of Guangxi Normal University(Natural Science Edition), 2025, 43(4): 58-68.

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