Adaptive Fourier Feature Physics-Informed Neural Networks for Nonlinear Cattaneo–Vernotte Heat Conduction with Temperature-Dependent Properties

Shirmohamadi, Shervin; Fallah, Aali; Mohammadi Aghdam, Mohammad

doi:10.22059/jcamech.2026.420632.1927

Adaptive Fourier Feature Physics-Informed Neural Networks for Nonlinear Cattaneo–Vernotte Heat Conduction with Temperature-Dependent Properties

Document Type : Research Paper

Authors

Shervin Shirmohamadi ¹

Aali Fallah ²

Mohammad Mohammadi Aghdam ¹

¹ Mechanical Engineering Department, Amirkabir University of Technology, Tehran 15875-4413, Iran

² Department of Automotive Engineering, Faculty of Engineering, Atilim University, 06830, Ankara, Türkiye

10.22059/jcamech.2026.420632.1927

Abstract

In this study, Physics-Informed Neural Networks (PINNs) are employed as mesh-free solvers for nonlinear non-Fourier heat conduction governed by the Cattaneo–Vernotte (CV) equation with temperature-dependent thermal conductivity. The CV formulation accounts for finite thermal relaxation effects and leads to a hyperbolic heat conduction model, for which analytical solutions are generally unavailable in the nonlinear regime. Three PINN architectures are investigated: a standard PINN (S-PINN), a Fourier Feature PINN (F-PINN), and a novel Adaptive Fourier Feature PINN (AF-PINN). The models are systematically evaluated in both linear and nonlinear regimes and validated against analytical and high-resolution numerical reference solutions. The results demonstrate that fixed Fourier feature mappings substantially improve accuracy compared to standard PINNs; however, their performance is highly sensitive to the choice of spectral bandwidth and may deteriorate under non-optimal initialization, particularly in nonlinear cases. The proposed AF-PINN overcomes this limitation by dynamically adapting the Fourier bandwidth during training, yielding robust convergence across a wide range of initial conditions. Statistical analyses based on multiple random initializations confirm consistently low prediction error and reduced variance. Further examination of the optimization dynamics highlights the critical role of decoupled learning rates for the spectral scale and network weights in enabling recovery from adverse high-frequency initializations. Overall, the adaptive framework enhances the accuracy, stability, and robustness of PINN-based solvers for nonlinear non-Fourier heat transfer and provides a reliable foundation for modeling wave-dominated and multiscale thermal phenomena.

Keywords

Nonlinear heat conduction

Physics-Informed Neural Networks

Fourier Feature PINNs

Adaptive spectral learning

Scientific Machine learning

Subjects

Computational material engineering

[1] A. Khosravirad, M. B. Ayani, Comparative analysis of thermal damage to laser-irradiated breast tumor based on Fourier conduction and non-Fourier heat conduction models: A numerical study, International Communications in Heat and Mass Transfer, Vol. 145, pp. 106837, 2023.

[2] D. Maillet, A review of the models using the Cattaneo and Vernotte hyperbolic heat equation and their experimental validation, International Journal of Thermal Sciences, Vol. 139, pp. 424-432, 2019.

[3] Y. Mao, Non-Fourier heat conduction of nano-films under ultra-fast laser, Materials, Vol. 16, No. 14, pp. 4988, 2023.

[4] M. M. Tung, Modeling the heating of biological tissue based on the hyperbolic heat transfer equation, Mathematical and Computer Modelling, Vol. 50, No. 5-6, pp. 665-672, 2009.

[5] C. Cattaneo, Sur une forme de l'equation de la chaleur eliminant la paradoxe d'une propagation instantantee, Compt. Rendu, Vol. 247, pp. 431-433, 1958.

[6] P. Vernotte, Les paradoxes de la theorie continue de l'equation de la chaleur, Comptes rendus, Vol. 246, pp. 3154, 1958.

[7] B. B. Xu, X. W. Gao, M. Cui, High precision simulation and analysis of non-Fourier heat transfer during laser processing, International Journal of Heat and Mass Transfer, Vol. 178, pp. 121574, 2021.

[8] S. Saedodin, M. J. Noroozi, D. D. Ganji, Nonlinear Analysis of a One-Dimensional Non-Fourier Heat Conduction Problem, Proceedings of the National Academy of Sciences, India Section A: Physical Sciences, Vol. 87, No. 3, pp. 447-454, 2017.

[9] R. Cai, C. Gou, H. Li, Algebraically explicit analytical solutions of unsteady 3-D nonlinear non-Fourier (hyperbolic) heat conduction, International Journal of Thermal Sciences, Vol. 45, No. 9, pp. 893-896, 2006.

[10] R. Shirmohammadi, A. Moosaie, Non-Fourier heat conduction in a hollow sphere with periodic surface heat flux, International Communications in Heat and Mass Transfer, Vol. 36, No. 8, pp. 827-833, 2009.

[11] S. C. Mishra, H. Sahai, Analyses of non-Fourier heat conduction in 1-D cylindrical and spherical geometry - An application of the lattice Boltzmann method, International Journal of Heat and Mass Transfer, Vol. 55, No. 23-24, pp. 7015-7023, 2012.

[12] H. Rahideh, P. Malekzadeh, M. R. Golbahar Haghighi, Non-Fourier Heat Conduction Analysis with Temperature-Dependent Thermal Conductivity, International Scholarly Research Notices, Vol. 2011, No. 1, pp. 321605, 2011.

[13] L. Wang, J. Xu, J. Wang, A peridynamic framework and simulation of non-Fourier and nonlocal heat conduction, International Journal of Heat and Mass Transfer, Vol. 118, pp. 1284-1292, 2018.

[14] W. Zhuoxin, Non-Fourier heat conduction analysis of a 2-D plate with inner cracks at arbitrary direction angles, Chinese Journal of Aeronautics, Vol. 35, No. 3, pp. 537-549, 2022.

[15] S. L. Brunton, J. N. Kutz, Methods for data-driven multiscale model discovery for materials, Journal of Physics: Materials, Vol. 2, No. 4, pp. 044002, 2019.

[16] S. L. Brunton, B. R. Noack, P. Koumoutsakos, Machine learning for fluid mechanics, Annual Review of Fluid Mechanics, Vol. 52, pp. 477-508, 2020.

[17] V. Yaghoubi, CNN-DST: ensemble deep learning based on Dempster-Shafer theory for vibration-based fault recognition, arXiv preprint arXiv:2110.07191, 2021.

[18] V. Yaghoubi, An ensemble classifier for vibration-based quality monitoring, Mechanical Systems and Signal Processing, Vol. 165, pp. 108341, 2022.

[19] H. Jin, E. Zhang, H. D. Espinosa, Recent advances and applications of machine learning in experimental solid mechanics: A review, Applied Mechanics Reviews, Vol. 75, No. 6, pp. 061001, 2023.

[20] J. Krzywanski, Heat and mass transfer prediction in fluidized beds of cooling and desalination systems by AI approach, Applied Thermal Engineering, Vol. 225, pp. 120200, 2023.

[21] A. Fallah, M. M. Aghdam, Physics-informed neural network for bending and free vibration analysis of three-dimensional functionally graded porous beam resting on elastic foundation, Engineering with Computers, Vol. 40, No. 1, pp. 437-454, 2024.

[22] M. Raissi, P. Perdikaris, G. E. Karniadakis, Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations, Journal of Computational Physics, Vol. 378, pp. 686-707, 2019.

[23] A. Fallah, M. M. Aghdam, Physics-Informed Neural Network for Solution of Nonlinear Differential Equations, in: R. N. Jazar, L. Dai, Nonlinear Approaches in Engineering Application: Automotive Engineering Problems, Eds., pp. 163-178, Cham: Springer Nature Switzerland, 2024.

[24] E. Haghighat, A physics-informed deep learning framework for inversion and surrogate modeling in solid mechanics, Computer Methods in Applied Mechanics and Engineering, Vol. 379, pp. 113741, 2021.

[25] L. Wu, N. G. Kilingar, L. Noels, A recurrent neural network-accelerated multi-scale model for elasto-plastic heterogeneous materials subjected to random cyclic and non-proportional loading paths, Computer Methods in Applied Mechanics and Engineering, Vol. 369, pp. 113234, 2020.

[26] S. S. Mirsadeghi Esfahani, A. Fallah, M. Mohammadi Aghdam, Physics-Informed Neural Network for Nonlinear Bending Analysis of Nano-Beams: A Systematic Hyperparameter Optimization, Mathematical and Computational Applications, Vol. 30, pp. 72, 2025.

[27] S. S. Mirsadeghi Esfahani, A. Fallah, M. Mohammadi Aghdam, Physics-informed Neural Network for Bending Analysis of Two-Dimensional Functionally Graded Nano-Beams based on Nonlocal Strain Gradient Theory, Journal of Computational Applied Mechanics, Vol. 56, No. 1, pp. 222-248, 2025.

[28] R. Nopour, A. Fallah, M. M. Aghdam, Large deflection analysis of functionally graded reinforced sandwich beams with auxetic core using physics-informed neural network, Mechanics Based Design of Structures and Machines, Vol. 53, No. 7, pp. 5264-5288, 2025.

[29] X. Zhuang, Deep autoencoder based energy method for the bending, vibration, and buckling analysis of Kirchhoff plates with transfer learning, European Journal of Mechanics - A/Solids, Vol. 87, pp. 104225, 2021.

[30] S. Cai, Physics-informed neural networks for heat transfer problems, Journal of Heat Transfer, Vol. 143, No. 6, pp. 060801, 2021.

[31] A. Hazra, P. Sarkar, S. Sarkar, PINN-based estimation of convective heat transfer in jet impingement cooling, Applied Thermal Engineering, pp. 129088, 2025.

[32] T. Lu, Gradient-Driven Physics Informed Neural Networks for Conduction Heat Transfer and Incompressible Laminar Flow, Journal of Computational and Nonlinear Dynamics, Vol. 21, No. 4, 2025.

[33] O. Fuks, H. A. Tchelepi, Limitations of physics informed machine learning for nonlinear two-phase transport in porous media, Vol. 1, No. 1, pp. 19-37, 2020.

[34] T. Zhang, Application of Fourier feature physics-information neural network in model of pipeline conveying fluid, Thin-Walled Structures, Vol. 198, pp. 111693, 2024.

[35] M. Tancik, Fourier features let networks learn high frequency functions in low dimensional domains, Advances in Neural Information Processing Systems, Vol. 33, pp. 7537-7547, 2020.

[36] K. Chandan, Predicting the thermal distribution in a convective wavy fin using a novel training physics-informed neural network method, Scientific Reports, Vol. 14, No. 1, pp. 7045, 2024.

[37] P. Cui, Solution and applications of parameterized convective heat transfer equations using physics-informed neural networks, International Journal of Heat and Mass Transfer, Vol. 247, pp. 127040, 2025.

[38] J. Lee, Extended multiphysics-informed neural network for conjugate heat transfer problems, International Journal of Heat and Mass Transfer, Vol. 246, pp. 127098, 2025.

[39] J. Z. Peng, Physics-informed graph convolutional neural network for modeling fluid flow and heat convection, Physics of Fluids, Vol. 35, No. 8, 2023.

[40] J. Xu, H. Wei, H. Bao, Physics-informed neural networks for studying heat transfer in porous media, International Journal of Heat and Mass Transfer, Vol. 217, pp. 124671, 2023.

[41] S. Shi, D. Liu, Z. Zhao, Non-Fourier Heat Conduction based on Self-Adaptive Weight Physics-Informed Neural Networks, in 2021 40th Chinese Control Conference (CCC), 2021.

[42] J. Zheng, F. Li, H. Huang, T-phPINN: Physics-informed neural networks for solving 2D non-Fourier heat conduction equations, International Journal of Heat and Mass Transfer, Vol. 235, pp. 126216, 2024.

[43] Q. A. Huhn, M. E. Tano, J. C. Ragusa, Physics-Informed Neural Network with Fourier Features for Radiation Transport in Heterogeneous Media, Nuclear Science and Engineering, Vol. 197, No. 9, pp. 2484-2497, 2023.

[44] O. Sallam, M. Fürth, On the use of Fourier Features-Physics Informed Neural Networks (FF-PINN) for forward and inverse fluid mechanics problems, Proceedings of the Institution of Mechanical Engineers, Part M: Journal of Engineering for the Maritime Environment, Vol. 237, No. 4, pp. 846-866, 2023.