三维稳态热传导方程参数反演问题的PINNs解法
PINNs Method for Parameter Inversion Problem of Three-Dimensional Steady-State Heat Conduction Equation
摘要: 热传导反问题广泛应用于地球科学、医学等领域，本文基于内嵌物理机理神经网络(PINNs)研究了一类三维有界域上热传导方程参数反演问题的数值解法。构建三维参数辨识模型，提出基于PINNs求解立方体区域三维稳态热传导方程参数反演问题的算法。通过同步反演热传导系数与整体温度值，实现热物性参数的高精度识别。在含噪声数据条件下，热传导系数反演的误差精度控制在10−4数量级，证实了算法对病态问题具有良好的抗噪性。
Abstract: Inverse heat conduction problem is widely used in earth science, medicine and other fields. In this paper, based on embedded physical mechanism neural network (PINNs), a numerical method for parameter inversion of heat conduction equation in three-dimensional bounded domain is studied. A three-dimensional parameter identification model is established, and an algorithm based on PINNs is proposed to solve the parameter inversion problem of three-dimensional steady-state heat conduction equation in cube region. Through simultaneous inversion of heat transfer coefficient and global temperature value, high precision identification of thermal physical property parameters is realized. Under the condition of noisy data, the error accuracy of heat transfer coefficient inversion is controlled at the order of 10−4, which proves that the algorithm has good anti-noise performance for ill-conditioned problems.
文章引用：阮晓晴, 邹琼. 三维稳态热传导方程参数反演问题的PINNs解法[J]. 应用数学进展, 2025, 14(5): 218-227. https://doi.org/10.12677/aam.2025.145250

参考文献

[1] 薛齐文, 魏伟. 非线性热传导反问题参数辨识[J]. 工程力学, 2010, 27(8): 5-10.
[2] 周焕林, 徐兴盛, 李秀丽, 等. 反演二维瞬态热传导问题随温度变化的导热系数[J]. 应用数学和力学, 2014, 35(12): 1341-1351.
[3] Mohebbi, F. and Sellier, M. (2016) Parameter Estimation in Heat Conduction Using a Two-Dimensional Inverse Analysis. International Journal for Computational Methods in Engineering Science and Mechanics, 17, 274-287. [Google Scholar] [CrossRef] 
[4] Yang, L. and Wang, Z. (2018) Artificial Neural Network (ANN) Modeling of Thermal Conductivity of Supercritical Ethane. Energy Sources, Part A: Recovery, Utilization, and Environmental Effects, 41, 396-404. [Google Scholar] [CrossRef] 
[5] 吴自库, 李福乐. 一维热传导方程热源反问题基于最小二乘法的正则化方法[J]. 计算物理, 2016, 33(1): 49-56.
[6] Haznedar, B. and Kalinli, A. (2018) Training ANFIS Structure Using Simulated Annealing Algorithm for Dynamic Systems Identification. Neurocomputing, 302, 66-74. [Google Scholar] [CrossRef] 
[7] Psichogios, D.C. and Ungar, L.H. (1992) A Hybrid Neural Network‐First Principles Approach to Process Modeling. AIChE Journal, 38, 1499-1511. [Google Scholar] [CrossRef] 
[8] Lagaris, I.E., Likas, A. and Fotiadis, D.I. (1998) Artificial Neural Networks for Solving Ordinary and Partial Differential Equations. IEEE Transactions on Neural Networks, 9, 987-1000. [Google Scholar] [CrossRef] [PubMed]
[9] Raissi, M., Perdikaris, P. and Karniadakis, G.E. (2019) Physics-Informed Neural Networks: A Deep Learning Framework for Solving Forward and Inverse Problems Involving Nonlinear Partial Differential Equations. Journal of Computational Physics, 378, 686-707. [Google Scholar] [CrossRef] 
[10] Cai, S., Wang, Z., Wang, S., Perdikaris, P. and Karniadakis, G.E. (2021) Physics-Informed Neural Networks for Heat Transfer Problems. Journal of Heat Transfer, 143, Article ID: 060801. [Google Scholar] [CrossRef] 
[11] Rasht‐Behesht, M., Huber, C., Shukla, K. and Karniadakis, G.E. (2022) Physics‐Informed Neural Networks (PINNs) for Wave Propagation and Full Waveform Inversions. Journal of Geophysical Research: Solid Earth, 127, e2021JB023120. [Google Scholar] [CrossRef] 
[12] 陈豪龙, 唐欣越, 柳兆涛, 等. 基于物理信息神经网络的非线性瞬态热传导正/反问题研究[J]. 重庆大学学报, 2024, 47(12): 124-136.
[13] Lu, L., Jin, P., Pang, G., Zhang, Z. and Karniadakis, G.E. (2021) Learning Nonlinear Operators via DeepONet Based on the Universal Approximation Theorem of Operators. Nature Machine Intelligence, 3, 218-229. [Google Scholar] [CrossRef] 
[14] Musgrave, J. and Huang, S.W. (2023) Fourier Domain Physics Informed Neural Network. arXiv: 2310.11459.
[15] Borate, P., Rivière, J., Marty, S., Marone, C., Kifer, D. and Shokouhi, P. (2024) Physics Informed Neural Network Can Retrieve Rate and State Friction Parameters from Acoustic Monitoring of Laboratory Stick-Slip Experiments. Scientific Reports, 14, Article No. 24624. [Google Scholar] [CrossRef] [PubMed]
[16] Zhang, H. and Liu, J. (2023) Solving an Inverse Source Problem by Deep Neural Network Method with Convergence and Error Analysis. Inverse Problems, 39, Article ID: 075013. [Google Scholar] [CrossRef] 
[17] Chanda, S., Balaji, C. and Venkateshan, S.P. (2018) Non-Intrusive Measurement of Thermal Contact Conductance at Polymer-Metal Two Dimensional Annular Interface. Heat and Mass Transfer, 55, 327-340. [Google Scholar] [CrossRef] 
[18] Liu, S. and Feng, L. (2020) An Inverse Problem for a Two-Dimensional Time-Fractional Sideways Heat Equation. Mathematical Problems in Engineering, 2020, Article ID: 5865971. [Google Scholar] [CrossRef] 
[19] Li, Y. and Hu, X. (2022) Artificial Neural Network Approximations of Cauchy Inverse Problem for Linear PDES. Applied Mathematics and Computation, 414, Article ID: 126678. [Google Scholar] [CrossRef] 
[20] 刘浩洋, 户将, 李勇锋, 等. 最优化: 建模、算法与理论[M]. 北京: 高教出版社, 2020.
[21] Xiang, Z., Peng, W., Liu, X. and Yao, W. (2022) Self-Adaptive Loss Balanced Physics-Informed Neural Networks. Neurocomputing, 496, 11-34. [Google Scholar] [CrossRef] 

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