基于局部网格的混合物理信息神经网络

孙久云, 董焕河, 方泳

系统科学与数学 ›› 2024, Vol. 44 ›› Issue (12) : 3760-3778.

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系统科学与数学 ›› 2024, Vol. 44 ›› Issue (12) : 3760-3778. DOI: 10.12341/jssms2023-0362

基于局部网格的混合物理信息神经网络

    孙久云, 董焕河, 方泳
作者信息 +

A New Hybrid Physics-Informed Neural Networks Based on Local Mesh

    SUN Jiuyun, DONG Huanhe, FANG Yong
Author information +
文章历史 +

摘要

文章提出了一种求解非线性偏微分方程的混合物理信息神经网络, (PINNs).在这个网络中,作者引入了基于局部网格的差分方法来构造物理残差并添加到损失函数中.因此, 混合 PINNs不完全依赖于自动微分技术, 且对解的梯度变化更敏感.此外, 由于PINNs是一种连续映射, 可以在任意时间和位置构造数值解,因此在求解域内任意点都能建立独立的局部网格. 同时, 由于网格是局部的,混合PINNs不会受到维度的限制. 最后, 通过数值实验验证了混合PINNs的性能, 并讨论了差分方法的阶数和局部网格的大小对解的精度的影响.实验结果表明, 混合PINNs的泛化能力明显优于 PINNs.

Abstract

In this paper, hybrid physics-informed neural networks(PINNs) are proposed for solving partial differential equations (PDEs). In this approach, we introduce a difference scheme based on local mesh to construct the physical residuals as part of the loss function. The obvious advantage is that the hybrid PINNs are not completely dependent on automatic differentiation techniques and are more sensitive to gradient changes in the solution. In addition, since the PINNs are continuous mappings, local meshes at arbitrary points can be built. Therefore, all local meshes are independent and the hybrid PINNs are not limited by dimension. Finally, the performance of the hybrid PINNs is verified by numerical experiments, and the effects of the order of differential schemes and the size of local mesh on accuracy are discussed. The results show that the generalization ability of the hybrid PINNs is more significant better than that of the PINNs.

关键词

非线性偏微分方程 / 物理信息神经网络 / 混合物理信息神经网络 / 数值解

Key words

Nonlinear partial differential equations / physics-informed neural networks / hybrid physics-informed neural networks / numerical solutions

引用本文

导出引用
孙久云 , 董焕河 , 方泳. 基于局部网格的混合物理信息神经网络. 系统科学与数学, 2024, 44(12): 3760-3778. https://doi.org/10.12341/jssms2023-0362
SUN Jiuyun , DONG Huanhe , FANG Yong. A New Hybrid Physics-Informed Neural Networks Based on Local Mesh. Journal of Systems Science and Mathematical Sciences, 2024, 44(12): 3760-3778 https://doi.org/10.12341/jssms2023-0362
中图分类号: 35Q51    35Q53    35Q55   

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基金

国家自然科学基金(11975143,12105161)资助课题.
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