Mc-DSOSLS: Mass Constraint-Based Deep Second-Order System Least Squares Method for Solving the Cahn-Hilliard Equation on Evolving Surfaces

Abstract

This paper employs the mass constraint-based deep second-order system least squares (Mc-DSOSLS) method to solve the Cahn-Hilliard (CH) equation on evolving surfaces. The CH equation models phase separation processes and presents significant computational challenges when extended to dynamic geometries such as manifolds. The proposed method utilizes the Mc-DSOSLS framework, which ensures numerical stability and accuracy by reformulating the problem as a second-order system and minimizing the associated least squares functional while conserving the total initial volume. A unique framework for generating the training dataset is presented, incorporating two novel algorithms. Initially, the Latin Hypercube (LH) sampling is applied to generate a uniformly distributed set of random points across the initial surface, ensuring smooth data coverage throughout the complex dynamics. As time progresses, the LH sampling is extended into the temporal direction, while a fresh spatial mesh is randomly regenerated at each time level to maintain variability and comprehensive coverage. The local generalization error bounds for the solution and the chemical potential are established. We validate the method through benchmark numerical experiments, demonstrating its effectiveness in capturing the evolving phase dynamics with high fidelity. The results confirm the stability and accuracy of the Mc-DSOSLS approach for solving the CH equation in complex evolving surface geometries.

Keywords

Cahn-Hilliard (CH) equation / Evolving surfaces / Mass-constraint / Deep learning / Error estimate / 35R01 / 35K55 / 65M15 / 65M75

References

[1] Archer AJ, Evans R. Dynamical density functional theory and its application to spinodal decomposition. J. Chem. Phys.. 2004, 121(9): 4246-4254.

[2] Armstrong NJ, Painter KJ, Sherratt JA. A continuum approach to modelling cell-cell adhesion. J. Theor. Biol.. 2006, 243(1): 98-113.

[3] Baumgart T, Hess ST, Webb WW. Imaging coexisting fluid domains in biomembrane models coupling curvature and line tension. Nature. 2003, 425(6960): 821-824.

[4] Bausch A, Bowick MJ, Cacciuto A, Dinsmore AD, Hsu MF, Nelson DR, Nikolaides MG, Travesset A, Weitz DA. Grain boundary scars and spherical crystallography. Science. 2003, 299: 1716-1718.

[5] Baydin AG, Pearlmutter BA, Radul AA, Siskind JM. Automatic differentiation in machine learning: a survey. J. Mach. Learn. Res.. 2018, 18: 1-43

[6] Beschle CA, Kovács B. Stability and error estimates for non-linear Cahn-Hilliard-type equations on evolving surfaces. Numer. Math.. 2022, 151(1): 1-48.

[7] Caflisch RE. Monte Carlo and quasi-Monte Carlo methods. Acta Numer.. 1998, 7: 1-49.

[8] Cahn JW, Hilliard JE. Free energy of a nonuniform system. I. Interfacial free energy. J. Chem. Phys.. 1958, 28(2): 258-267.

[9] Du Q, Ju L, Tian L. Finite element approximation of the Cahn-Hilliard equation on surfaces. Comput. Methods Appl. Mech. Eng.. 2011, 200: 2458-2470.

[10] Dziuk G, Elliott CM. Finite elements on evolving surfaces. IMA J. Numer. Anal.. 2007, 27(2): 262-292.

[11] Eilks C, Elliott C. Numerical simulation of dealloying by surface dissolution via the evolving surface finite element method. J. Comput. Phys.. 2008, 227: 9727-9741.

[12] Elliott C, Stinner B. A surface phase field model for two-phase biological membranes. SIAM J. Appl. Math.. 2010, 70(8): 2904-2928.

[13] Elliott C, Stinner B. Modeling and computation of two phase geometric biomembranes using surface finite elements. J. Comput. Phys.. 2010, 229: 6585-6612.

[14] Elliott CM, Ranner T. Evolving surface finite element method for the Cahn-Hilliard equation. Numer. Math.. 2015, 129(3): 483-534.

[15] Erlebacher J, Aziz MJ, Karma A, Dimitrov N, Sieradzki K. Evolution of nanoporosity in dealloying. Nature. 2001, 410(6827): 450-453.

[16] Gao W, Wang C. Active learning based sampling for high-dimensional nonlinear partial differential equations. J. Comput. Phys.. 2023, 475: 111848.

[17] Garcke H, Lam K, Stinner B. Diffuse interface modelling of soluble surfactants in two-phase flow. Commun. Math. Sci.. 2014, 121475-1522.

[18] Hohenberg P, Halperin B. Theory of dynamic critical phenomena. Rev. Mod. Phys.. 1977, 49: 435-479.

[19] Hornik K, Stinchcombe M, White H. Multilayer feedforward networks are universal approximations. Neural Netw.. 1989, 2(5): 359-366.

[20] Kim J. Phase-field models for multi-component fluid flows. Commun. Comput. Phys.. 2012, 12(3): 613-661.

[21] Kingma, D.P., Adam, J.B.: A method for stochastic optimization. arXiv: 1412.6980 (2014)

[22] Li Y, Choi JI, Kim J. A phase-field fluid modeling and computation with interfacial profile correction term. Commun. Nonlinear Sci. Numer. Simul.. 2016, 30: 84-100.

[23] Li Y, Qi X, Kim J. Direct discretization method for the Cahn-Hilliard equation on an evolving surface. J. Sci. Comput.. 2018, 77: 1147-1163.

[24] Mercker M, Ptashnyk M, Kuhnle J, Hartmann D, Weiss M, Jager W. A multiscale approach to curvature modulated sorting in biological membranes. J. Theor. Biol.. 2012, 301: 67-82.

[25] Mishra S, Molinaro R. Estimates on the generalization error of physics-informed neural networks for approximating PDEs. IMA J. Numer. Anal.. 2023, 43(1): 1-43.

[26] O’Connor, D., Stinner, B.: The Cahn-Hilliard equation on an evolving surface. arXiv:1607.05627 (2016)

[27] Raissi M, Perdikaris P, Karniadakis GE. Physics-informed neural networks: a deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations. J. Comput. Phys.. 2019, 378: 686-707.

[28] Tang P, Qiu F, Zhang H, Yang Y. Phase separation patterns for diblock copolymers on spherical surfaces: a finite volume method. Phys. Rev. E Stat. Nonlinear Soft Matter Phys.. 2005, 72: 016710.

[29] Wise S, Lowengrub J, Frieboes H, Cristini V. Three-dimensional multispecies nonlinear tumor growth-I: model and numerical method. J. Theor. Biol.. 2008, 253524-543.

[30] Yang SD, Lee HG, Kim J. A phase-field approach for minimizing the area of triply periodic surfaces with volume constraint. Comput. Phys. Commun.. 2010, 181(6): 1037-1046.

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Shanghai University