CONSTRAINED NEURAL NETWORKS APPROACH OF THE CAHN-HILLIARD EQUATION WITH $u$-DEPENDENT MOBILITY
Keywords:
Cahn-Hilliard, phase separation, concentration-dependent mobility, c-PINNs, finite element method, neural networks, physical constraintsDOI:
https://doi.org/10.17654/0975045226003Abstract
This paper presents a comparative study of numerical methods for the Cahn-Hilliard equation with concentration dependent mobility, contrasting a classical finite element approach with a modern deep learning framework. We introduce a constrained Physics-Informed Neural Network (c-PINNs), which directly incorporates the physical bounds of the concentration field into the learning process. Our results demonstrate that this approach not only captures the complex dynamics of phase separation with high visual fidelity but also achieves strong quantitative agreement with a high fidelity finite element reference solution. The c-PINNs method adeptly handles the high order spatial derivatives inherent to the problem via automatic differentiation, bypassing the complexities of traditional discretization. While the initial training phase is computationally intensive, the resulting model acts as a fast and accurate surrogate, capable of instantaneous predictions. This work highlights the potential of constrained neural networks as a robust and flexible alternative for simulating complex physical phenomena, paving the way for more efficient exploration of phase separation dynamics in materials science and beyond.
Received: August 3, 2025
Revised: August 9, 2025
Accepted: September 19, 2025
References
[1] M. Abdel-Aty, D. Abo-Kahla and A. S. F. Obada, Spatial dependence of moving three-level atoms interacting with a three-laser beam, Canadian Journal of Physics 91(12) (2013), 1068-1073.
[2] D. A. M. Abo-Kahla, The Pancharatnam phase of a three-level atom coupled to two systems of n-two level atoms, Journal of Quantum Information Science 1(1) (2011), 44-55.
[3] D. A. M. Abo-Kahla, The atomic inversion and the purity of a quantum dot two-level systems, Applied Mathematics and Information Sciences 10(4) (2016), 1-5.
[4] D. A. M. Abo-Kahla, Long-lived quantum coherence and nonlinear properties of a two-dimensional semiconductor quantum well, Journal of the Optical Society of America B 37(11) (2020), A96¬-A109.
[5] D. A. M. Abo-Kahla, Long-lived quantum coherence in a two-level semiconductor quantum dot, Pramana 94(1) (2020), 1-14.
[6] N. D. Alikakos, P. W. Bates and G. Fusco, On the Cahn-Hilliard equation with concentration dependent mobility, J. Differential Equations 90 (2017), 81-135.
[7] John Barrett, James Blowey and Harald Garcke, Finite element approximation of the Cahn-Hilliard equation with degenerate mobility, SIAM J. Numer. Anal. 37(1) (2000), 286-318.
[8] Atilim Gunes Baydin, Barak A. Pearlmutter, Alexey Andreyevich Radul and Jeffrey Mark Siskind, Automatic differentiation in machine learning: a survey, J. Mach. Learn. Res. 18 (2018), 1-43.
[9] N.-E. Bohne, Patrick Ciarlet, Jr. and S. Sauter, Crouzeix-Raviart elements on simplicial meshes in d dimensions, 2024. arXiv:2407.04361 DOI: 10.48550/arXiv.2407.04361.
[10] J. W. Cahn, Phase separation by Spinodal decomposition in alloys, Acta Metallurgica 9(9) (1961), 795-801.
[11] X. Chen, Y. Cheng, J. Xu and C. Wang, Cahn-Hilliard inpainting and a generalization for gray-value images, SIAM J. Appl. Math. 83 (2023), 123-145.
[12] Craig Cowan. The Cahn-Hilliard equation as a gradient flow, PhD thesis, Simon Fraser University, 2005. researchgate.net/publication/241623162.
[13] M. Crouzeix and P. A. Raviart, Conforming and nonconforming finite element methods for solving the stationary stokes equations, Mathematical Modelling and Numerical Analysis 7(R3) (1973), 33-75.
[14] C. M. Elliott and J. W. Cahn, The Cahn-Hilliard equation with a concentration dependent mobility: motion by minus the Laplacian of the mean curvature, European J. Appl. Math. 7(3) (1996), 287-301.
[15] Charles M. Elliott and Stig Larsson, A second order error estimate for a time-dependent Cahn-Hilliard equation, Mathematical Modelling and Numerical Analysis 23(3) (1989), 397-405.
[16] David J. Eyre, Unconditionally gradient stable time-marching the Cahn-Hilliard equation, MRS Online Proceedings Library (OPL) 529 (1998), 39-46.
[17] Hanzhong Gao, Lu Sun and J. X. Wang, Phygeonet: physics-informed geometry adaptive convolutional neural networks for solving parametric pdes on irregular domain, J. Comput. Phys. 428 (2021), 110079.
[18] Souvik Goswami, Minglang Yin, Yue Yu and George Em Karniadakis, Physics-informed neural networks for surrogate modeling of parametric solutions to the Allen-Cahn and Cahn-Hilliard equations, Computational Mechanics 69(4) (2022), 957-981.
[19] Christopher P. Grant, Spinodal decomposition for the Cahn-Hilliard equation, Comm. Partial Differential Equations 18(3-4) (1993), 453-490.
[20] G. Grun, Existence of solutions for the Cahn-Hilliard equation with degenerate mobility, Nonlinear Analysis: Theory, Methods and Applications 25(8) (1995), 1001-1010.
[21] J. E. Hilliard, The theory of phase separation in alloys, Journal of Chemical Physics 53(8) (1970), 2924-2933.
[22] Junseok Kim, Seunggyu Lee, Yongho Choi, Seok-Min Lee and Darae Jeong, Basic principles and practical applications of the Cahn-Hilliard equation, Math. Probl. Eng. 2016(1) (2016), 9532608.
[23] M. Lupu, A. Miranville, A. Signori, C. Walker and H. Wu, Cahn-Hilliard models for medical imaging: a review, Journal of Biomedical Imaging 2021, Article ID 6666354, pp. 1-15.
[24] A. Miranville,A. Signori and S. Zelik, Image prediction through Cahn-Hilliard image inpainting, Royal Society Open Science 8 (2021), 201294
[25] Flore Nabet, Numerical analysis for generalized Cahn-Hilliard type equations with applications to tumor growth, PhD thesis, École Polytechnique, 2023
[26] Morteza A. Nabian, Mohammad Zaki and George E. Karniadakis, Efficient physics informed neural networks for solving forward and inverse problems involving nonlinear partial differential equations, Computer Methods in Applied Mechanics and Engineering 384 (2021), 113938.
[27] A. A. Nahla, D. A. M. Abo-Kahla and M. H. Raddadi, Entanglement and atomic inversion for two two-level atoms interacting with an intensity-dependent quantized field, Internat. J. Modern Phys. B 36(26) (2022), 2250179.
[28] A. Novick-Cohen, Energy methods for the Cahn-Hilliard equation, Nonlinear Diffusion Equations and their Applications, Springer, 1985, pp. 1-20.
[29] M. H. Raddadi, A. A. Nahla and D. A. M. Abo-Kahla, Quantized study for asymmetric two two-level atoms interacting with intensity-dependent coupling regime, Indian Journal of Physics 97(5) (2023), 1345-1358.
[30] Maziar Raissi, Paris Perdikaris and George E. Karniadakis, Physics-informed neural networks: a deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations, J. Comput. Phys. 391 (2019), 686-707.
[31] Jie Shen and Xiaofeng Yang, Numerical methods for the Cahn-Hilliard equation, Discrete and Continuous Dynamical Systems 28(4) (2010), 1649.
[32] Hao Wu, A review on the Cahn-Hilliard equation: Classical results and recent advances in dynamic boundary conditions, Discrete and Continuous Dynamical Systems - Series S 14(1) (2021), 1-54.
[33] W. Yin, Existence of solutions for the Cahn-Hilliard equation with degenerate mobility in one dimension, J. Differential Equations 97(1) (1992), 1-16.
[34] Y. Zhang , B. Liu and Z. Chen, Fast image inpainting strategy based on the space-fractional modified Cahn-Hilliard equations, Comput. Math. Appl. 81 (2020), 1234-1245.
Downloads
Published
Issue
Section
License
Copyright (c) 2025 PUSHPA PUBLISHING HOUSE, PRAYAGRAJ, INDIA
This work is licensed under a Creative Commons Attribution 4.0 International License.
Attribution: Credit Pushpa Publishing House as the original publisher, including title and author(s) if applicable.
Non-Commercial Use: For non-commercial purposes only. No commercial activities without explicit permission.
No Derivatives: Modifying or creating derivative works not allowed without written permission.
Contact Pushpa Publishing House for more info or permissions.
Publication count: 
Citation count: 
Google h-index: 
Downloads: 