基于内嵌物理机理神经网络的热传导方程的正问题及逆问题求解

赵暾; 周宇; 程艳青; 钱炜祺

doi:10.7638/kqdlxxb-2020.0176

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基于内嵌物理机理神经网络的热传导方程的正问题及逆问题求解

研究论文 | 更新时间：2026-05-28

- 基于内嵌物理机理神经网络的热传导方程的正问题及逆问题求解
- Solving forward and inverse problems of the heat conduction equation using physics-informed neural networks
- 空气动力学学报 2021年39卷第5期页码：19-26
- 作者机构：
  
  1. 空气动力学国家重点实验室,绵阳,621000
  2. 中国空气动力研究与发展中心,绵阳,621000
- 作者简介：
  
  赵暾（1989-），男，甘肃西和人，助理研究员，研究方向：飞行仿真与智能控制. E-mail：zhaotun@cardc.cn
  钱炜祺*（1973-），男，江苏无锡人，研究员，研究方向：空气动力学与气动参数辨识. E-mail：qwqhyy@sina.com
- 基金信息：
- DOI：10.7638/kqdlxxb-2020.0176
  中图分类号： V231.1+3
- 纸质出版：2021
- 稿件说明：
移动端阅览
赵暾, 周宇, 程艳青, 等. 基于内嵌物理机理神经网络的热传导方程的正问题及逆问题求解[J]. 空气动力学学报, 2021, 39(5): 19−26.

ZHAO Tun, ZHOU Yu, CHENG Yanqing, et al. Solving forward and inverse problems of the heat conduction equation using physics-informed neural networks[J]. 2021, 39(5): 19-26.
赵暾, 周宇, 程艳青, 等. 基于内嵌物理机理神经网络的热传导方程的正问题及逆问题求解[J]. 空气动力学学报, 2021, 39(5): 19−26. DOI： 10.7638/kqdlxxb-2020.0176

ZHAO Tun, ZHOU Yu, CHENG Yanqing, et al. Solving forward and inverse problems of the heat conduction equation using physics-informed neural networks[J]. 2021, 39(5): 19-26. DOI： 10.7638/kqdlxxb-2020.0176.

摘要

建立了一种基于内嵌物理机理神经网络（PINN）的热传导方程的正问题及逆问题求解方法。该方法利用自动微分技术将一维热传导方程嵌入到深度网络的损失函数中，通过以损失函数最小为目标来优化深度网络，求解一维热传导方程以及对方程中的未知导热系数进行辨识。随后，分析了基于PINN求解正问题的收敛精度以及参数辨识的鲁棒性，并得出以下结论：在给定网络结构的情况下，基于PINN求解一维热传导方程的收敛误差在样本点数较少时主要由采样误差主导，而当样本点数较多时，收敛误差由优化误差主导；由于损失函数中包含了方程相关的正则化项，以及采用了自动微分技术，因此，基于PINN的参数辨识方法噪声标签数据具有较强的鲁棒性。

Abstract

A method for solving the forward and inverse problems of the heat conduction equation based on a physics-informed neural network (PINN) is proposed. In this method

the one-dimensional heat conduction equation is integrated into a loss function as the regularization term using the automatic differentiation. Consequently

the solution of the equation as well as the embedded unknown physical parameters can both be obtained by minimizing the loss function of the PINN. The convergence errors of solving the forward problem and the robustness of parameter identification are analyzed. Results show that for a given network structure

the convergence errors of solving the one-dimensional heat conduction equation based on PINN are is mainly determined by the sampling errors when sampling points are insufficient

while for abundant samples

the convergence error is determined by the optimization errors. Since the loss function contains equation-related regularization terms and adopts the automatic differentiation technique

the identification method based on PINN is robust to noisy data.

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references

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