热传导方程两类参数的统计反演方法
Statistical Inversion Methods for Two Types of Parameters in the Heat Conduction Equation
摘要: 本文以热传导方程为背景，分别研究了不同位置上的常数型未知参数与函数型未知参数，并给出统计反演算法，完成数值反演试验。两类方法均能高效完成反演任务，且在面对高噪声观测信息时，具有很强的鲁棒性。
Abstract: Focusing on the heat conduction equation, this paper investigates constant-type unknown parameters and function-type unknown parameters at different positions, and provides statistical inversion algorithms to complete numerical inversion experiments. Both methods can efficiently accomplish the inversion tasks and exhibit strong robustness in the presence of high-noise observational information.
文章引用：胥文博. 热传导方程两类参数的统计反演方法[J]. 应用数学进展, 2025, 14(3): 203-215. https://doi.org/10.12677/aam.2025.143106

参考文献

[1] Yang, L., Deng, Z., Yu, J. and Luo, G. (2009) Optimization Method for the Inverse Problem of Reconstructing the Source Term in a Parabolic Equation. Mathematics and Computers in Simulation, 80, 314-326. [Google Scholar] [CrossRef] 
[2] Wei, T. and Zhang, Z.Q. (2013) Reconstruction of a Time-Dependent Source Term in a Time-Fractional Diffusion Equation. Engineering Analysis with Boundary Elements, 37, 23-31. [Google Scholar] [CrossRef] 
[3] Wang, J. and Zabaras, N. (2004) A Bayesian Inference Approach to the Inverse Heat Conduction Problem. International Journal of Heat and Mass Transfer, 47, 3927-3941. [Google Scholar] [CrossRef] 
[4] 吴自库, 李福乐, Kwak, D.Y. 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): 49-55.
[5] 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] 
[6] 陆金甫, 关治. 偏微分方程数值解法[M]. 第2版. 北京: 清华大学出版社, 2004: 82-120.
[7] Peaceman, D.W. and Rachford Jr., H.H. (1955) The Numerical Solution of Parabolic and Elliptic Differential Equations. Journal of the Society for Industrial and Applied Mathematics, 3, 28-41. [Google Scholar] [CrossRef] 
[8] Kaipo, J., Somersalo, E. 统计与计算反问题[M]. 刘逸侃, 译. 北京: 科学出版社, 2018: 13-43.
[9] Yan, L., Yang, F. and Fu, C. (2009) A Bayesian Inference Approach to Identify a Robin Coefficient in One-Dimensional Parabolic Problems. Journal of Computational and Applied Mathematics, 231, 840-850. [Google Scholar] [CrossRef] 
[10] 刘艳杰. 参数反演的贝叶斯方法及其应用研究[D]: [硕士学位论文]. 淄博: 山东理工大学, 2020.
[11] 张国安. 一类二维多层对流扩散方程参数反演的神经网络方法研究[D]: [硕士学位论文]. 上海: 上海财经大学, 2022.
[12] 刘云美, 史正梅. PINNs在反演计算中影响因素的数值比较分析[J]. 新乡学院学报, 2024, 41(9): 14-21.
[13] Jagtap, A.D., Kawaguchi, K. and Karniadakis, G.E. (2020) Adaptive Activation Functions Accelerate Convergence in Deep and Physics-Informed Neural Networks. Journal of Computational Physics, 404, Article ID: 109136. [Google Scholar] [CrossRef] 
[14] Jagtap, A.D., Kawaguchi, K. and Em Karniadakis, G. (2020) Locally Adaptive Activation Functions with Slope Recovery for Deep and Physics-Informed Neural Networks. Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences, 476, Article ID: 20200334. [Google Scholar] [CrossRef] [PubMed]
[15] Deng, Z., Yang, L., Yu, J. and Luo, G. (2013) Identifying the Diffusion Coefficient by Optimization from the Final Observation. Applied Mathematics and Computation, 219, 4410-4422. [Google Scholar] [CrossRef] 

为你推荐