Physics-informed neural networks for solving steady-state heat conduction inverse problems

Tangwei Liu; Qiong  Zou; Xiaoqing  Ruan; Dingding Yan; Jeevan Kafle; Zhongzhou Lan

doi:10.59400/adecp3584

Physics-informed neural networks for solving steady-state heat conduction inverse problems

Tangwei Liu
School of Science, East China University of Technology, Nanchang 330013, China
Qiong Zou
School of Science, East China University of Technology, Nanchang 330013, China
Xiaoqing Ruan
School of Science, East China University of Technology, Nanchang 330013, China; Mathematics Teaching and Research Group, Sheshan School, Shanghai 201602, China
Dingding Yan
School of Science, East China University of Technology, Nanchang 330013, China
Jeevan Kafle
Central Department of Mathematics, Tribhuvan University, Kathmandu 44600, Nepal
Zhongzhou Lan
School of Computer Information Management, Inner Mongolia University of Finance and Economics, Hohhot 010070, China

Article ID: 3584

Keywords: physics-informed neural networks; three-dimensional heat conduction; inverse problem; loss function; numerical simulation

Abstract

This paper investigates numerical methods for a class of three-dimensional steady-state heat conduction inverse problems. By employing Physics-Informed Neural Networks (PINNs), the three-dimensional heat conduction inverse problems are reformulated as optimization problems with respect to a properly defined loss function. Two cases with different additional conditions are considered: one incorporates an additional boundary temperature gradient condition, while the other involves an additional partial internal temperature measurement. Corresponding efficient algorithms are developed to solve the resulting optimization problems. To optimize the performance of the proposed numerical framework, systematic sensitivity analyses are performed to rigorously justify the selection of key hyperparameters (e.g., activation functions and network architecture). Additionally, to validate the mathematical effectiveness and noise robustness of the algorithm, this study primarily employs synthetic data with controlled noise levels for quantitative evaluation. Numerical results demonstrate that the proposed method can efficiently and accurately approximate the solutions to the three-dimensional (3D) heat conduction inverse problems for both polynomial and non-polynomial cases. In addition, a theoretical analysis is provided to interpret the method's stability against noise amplification. Future work will focus on applying the proposed framework to real-world field data to further validate its practical engineering value.

Published

2026-03-05

How to Cite

Liu, T., Zou, Q., Ruan, X., Yan, D., Kafle, J., & Lan, Z. (2026). Physics-informed neural networks for solving steady-state heat conduction inverse problems. Advances in Differential Equations and Control Processes, 33(1). https://doi.org/10.59400/adecp3584

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Issue

Vol. 33 No. 1 (2026)

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Article

This work is licensed under a Creative Commons Attribution 4.0 International License.

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