Skip to main content
SpringerLink
Log in

Fractional physics-informed neural networks for time-fractional phase field models

  • Original Paper
  • Published:
Nonlinear Dynamics Aims and scope Submit manuscript

Abstract

In this paper, a new fractional physics-informed neural networks (fPINNs) is proposed, which combines fPINNs with spectral collocation method to solve the time-fractional phase field models. Compared to fPINNs, it has large representation capacity due to the property of spectral collocation method, which reduces the number of approximate points of discrete fractional operators, improves the training efficiency and has higher error accuracy. Unlike the traditional numerical method, it directly optimizes the spectral collocation coefficient, saves the time of matrix calculation, is easy to deal with the high-dimensional model, and also has higher error accuracy. First, fPINNs based on a spectral collocation method is used to obtain the numerical solutions of the models under consideration. The spectral collocation method is used to discretize the space direction, and the fractional backward difference formula is used to approximate the time-fractional derivative. The error accuracy in different cases is discussed, and it is observed that the point-wise error is \(10^{-5}\) to \(10^{-7}\). Next, fPINNs is employed to solve several inverse problems in time-fractional phase field models to identify the order of fractional derivative, mobility constant, and other coefficients. The results of numerical experiments are presented to prove the effectiveness of fPINNs in solving time-fractional phase field models and their inverse problems.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9
Fig. 10
Fig. 11
Fig. 12
Fig. 13
Fig. 14
Fig. 15
Fig. 16
Fig. 17
Fig. 18
Fig. 19
Fig. 20
Fig. 21
Fig. 22

Similar content being viewed by others

Data availability statement

The datasets generated and analyzed during the current study are available from the corresponding author on reasonable request.

References

  1. Brunton, S., Proctor, J., Kutz, J.: Discovering governing equations from data by sparse identification of nonlinear dynamical systems. Proc. Nat. Acad. Sci. 113(15), 3932–3937 (2016)

    MathSciNet  MATH  Google Scholar 

  2. Brunton, S., Noack, B., Koumoutsakos, P.: Machine learning for fluid mechanics. Annu. Rev. Fluid Mech. 52, 477–508 (2020)

    MATH  Google Scholar 

  3. Gao, G., Li, J., Wen, Y.: DeepComfort: energy-efficient thermal comfort control in buildings via reinforcement learning. IEEE Internet Things J. 7, 8472–8484 (2020)

    Google Scholar 

  4. Bouman, K., Xiao, B., Battaglia, P., Freeman, W.: Estimating the material properties of fabric from video. In: Proceedings of the IEEE International Conference on Computer Vision. 1984–1991 (2013)

  5. Zhao, R., Yan, R., Chen, Z., Mao, K., Wang, P., Gao, R.: Deep learning and its applications to machine health monitoring. Mech. Syst. Signal Process. 115, 213–237 (2019)

    Google Scholar 

  6. Wang, Q., Zhang, G., Sun, C., Wu, N.: High efficient load paths analysis with U index generated by deep learning. Comput. Methods Appl. Mech. Eng. 344, 499–511 (2019)

    MathSciNet  MATH  Google Scholar 

  7. Finol, D., Lu, Y., Mahadevan, V., Srivastava, A.: Deep convolutional neural networks for eigenvalue problems in mechanics. Internat. J. Numer. Methods Eng. 118(5), 258–275 (2019)

    MathSciNet  Google Scholar 

  8. Karpatne, A., Atluri, G., Faghmous, J., Steinbach, M., Banerjee, A., Ganguly, A., Shekhar, S., Samatova, N., Kumar, V.: Theory-guided data science: a new paradigm for scientific discovery from data. IEEE Trans. Knowl. Data Eng. 29(10), 2318–2331 (2017)

    Google Scholar 

  9. Raissi, M., Perdikaris, P., Karniadakis, G.: Physics-informed neural networks: a deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations. J. Comput. Phys. 378, 686–707 (2019)

    MathSciNet  MATH  Google Scholar 

  10. Kharazmi, E., Zhang, Z., Karniadakis, G.: hp-VPINNs: variational physics-informed neural networks with domain decomposition. Comput. Methods Appl. Mech. Eng. 374(C), 113547 (2021)

    MathSciNet  MATH  Google Scholar 

  11. Pang, G., D’Elia, M., Parks, M., Karniadakis, G.: nPINNs: nonlocal physics-informed neural networks for a parametrized nonlocal universal Laplacian operator. Algorithms Appl. J. Comput. Phys. 422, 109760 (2020)

    MathSciNet  MATH  Google Scholar 

  12. Jagtap, A., Karniadakis, G.: Extended physics-informed neural networks (XPINNs): a generalized space-time domain decomposition based deep learning framework for nonlinear partial differential equations. Commun. Comput. Phys. 28(5), 2002–2041 (2020)

    MathSciNet  MATH  Google Scholar 

  13. Zheng, Q., Zeng, L., Karniadakis, G.: Physics-informed semantic inpainting: application to geostatistical modeling. J. Comput. Phys. 419, 109676 (2020)

    MathSciNet  MATH  Google Scholar 

  14. Giusti, A.: A comment on some new definitions of fractional derivative. Nonlinear Dyn. 93, 1757–1763 (2018)

    MATH  Google Scholar 

  15. Wu, G., Baleanu, D.: Discrete fractional logistic map and its chaos. Nonlinear Dyn. 75, 283–287 (2014)

    MathSciNet  MATH  Google Scholar 

  16. Benson, D., Wheatcraft, S., Meerschaert, M.: Application of a fractional advection-dispersion equation. Water Resour. Res. 36, 1403–1412 (2000)

    Google Scholar 

  17. Sun, H., Zhang, Y., Chen, W., Reeves, D.: Use of a variable-index fractional-derivative model to capture transient dispersion in heterogeneous media. J. Contam. Hydrol. 157, 47–58 (2014)

    Google Scholar 

  18. Sun, H., Zhang, Y., Chen, W., Reeves, D.: Use of a variable-index fractional-derivative model to capture transient dispersion in heterogeneous media. J. Contam. Hydrol. 157, 47–58 (2014)

    Google Scholar 

  19. Chen, W., Holm, S.: Fractional laplacian time-space models for linear and nonlinear lossy media exhibiting arbitrary frequency power-law dependency. J. Acous. Soc. Amer. 115, 1424–1430 (2004)

    Google Scholar 

  20. Chen, W.: A speculative study of 2/3-order fractional laplacian modeling of turbulence: some thoughts and conjectures. Chaos 16, 023126 (2006)

    MATH  Google Scholar 

  21. Sirignano, J., Spiliopoulos, K.: DGM: a deep learning algorithm for solving partial differential equations. J. Comput. Phys. 375, 1339–1364 (2018)

    MathSciNet  MATH  Google Scholar 

  22. Berg, J., Nystr\(\ddot{o}\)m, K.: A unified deep artificial neural network approach to partial differential equations in complex geometries. Neurocomputing 317, 28–41 (2018)

  23. Meng, X., Li, Z., Zhang, D., Karniadakis, G.: PPINN: parareal physics-informed neural network for time-dependent PDEs. Comput. Methods Appl. Mech. Eng. 370, 113250 (2020)

    MathSciNet  MATH  Google Scholar 

  24. Pang, G., Lu, L., Karniadakis, G.: fPINNs: fractional physics informed neural networks. SIAM J. Sci. Comput. 41, 2603–2626 (2019)

    MathSciNet  MATH  Google Scholar 

  25. Rostami, F., Jafarian, A.: A new artificial neural network structure for solving high-order linear fractional differential equations. Int. J. Comput. Math. 95(3), 528–539 (2018)

    MathSciNet  MATH  Google Scholar 

  26. Hajimohammadi, Z., Baharifard, F., Ghodsi, A.: Fractional chebyshev deep neural network (FCDNN) for solving differential models. Chaos Solitons Fractals 153(2), 111530 (2021)

    MathSciNet  MATH  Google Scholar 

  27. Qu, H., Liu, X., She, Z.: Neural network method for fractional-order partial differential equations. Neurocomputing 414, 225–237 (2020)

    Google Scholar 

  28. Rasanan, A., Bajalan, N., Parand, K., Rad, J.: Simulation of nonlinear fractional dynamics arising in the modeling of cognitive decision making using a new fractional neural network. Math. Meth. Appl. Sci. 43(3), 1437–1466 (2020)

    MathSciNet  MATH  Google Scholar 

  29. Kapustina, M., Tsygankov, D., Zhao, J., Wesller, T., Yang, X., Chen, A., Roach, N., Elston, T., Wang, Q., Jacobson, K., Forest, M.: Modeling the excess cell membrane stored in a complex morphology of bleb-like protrusions. Plos Comput. Biol. 12(3), 1004841 (2016)

    Google Scholar 

  30. Shao, D., Pappel, W., Levine, H.: Computational model for cell morphodynamics. Phys. Rev. Lett. 105(10), 108104 (2010)

    Google Scholar 

  31. Ziebert, F., Aranson, I.: Effects of adhesion dynamics and substrate compliance on the shape and motility of crawling cells. Plos One 8(5), 64511 (2013)

    Google Scholar 

  32. Zhao, J., Shen, Y., Happasalo, M., Wang, Z., Wang, Q.: A 3D numerical study of antimicrobial persistence in heterogeneous multi-species biofilms. J. Theor. Biol. 392, 83–98 (2016)

    MathSciNet  MATH  Google Scholar 

  33. Kohn, R., Yan, X.: Upper bound on the coarsening rate for an epitaxial growth model. Comm. Pure Appl. Math. 56(11), 1549–1564 (2003)

    MathSciNet  MATH  Google Scholar 

  34. Li, B., Liu, J.: Epitaxial growth without slope selection: energetics, coarsening and dynamic scaling. J. Nonlinear Sci. 14(5), 429–451 (2004)

    MathSciNet  MATH  Google Scholar 

  35. Allen, S., Cahn, J.: A microscopic theory for antiphase boundary motion and its application to antiphase domain coarsening. Acta Metall. Mater. 27(6), 1085–1095 (1979)

    Google Scholar 

  36. Doi, M., Edwards, S.: The Theory of Polymer Dynamics, p. 73. Oxford University Press, Oxford (1988)

    Google Scholar 

  37. Cahn, J., Hilliard, J.: Free Energy of a Nonuniform System. I. Interfacial Free Energy. J. Chem. Phys. 28, 258–267 (1958)

    MATH  Google Scholar 

  38. Yue, P., Feng, J., Liu, C., Shen, J.: A diffuse-interface method for simulating two-phase flows of complex fluids. J. Fluid Mech. 515, 293–317 (2004)

    MathSciNet  MATH  Google Scholar 

  39. Anderson, D., Mcfadden, G., Wheeler, A.: Diffuse-interface methods in fluid mechanics. Annu. Rev. Fluid Mech. 30, 139–165 (1997)

    MathSciNet  MATH  Google Scholar 

  40. Ainsworth, M., Mao, Z.: Analysis and approximation of a fractional Cahn–Hilliard equation. SIAM J. Numer. Anal. 55(4), 1689–1718 (2017)

    MathSciNet  MATH  Google Scholar 

  41. Ainsworth, M., Mao, Z.: Well-posedness of the Cahn–Hilliard equation with fractional free energy and its Fourier Galerkin approximation. Chaos Soliton. Fract. 102, 264–273 (2017)

    MathSciNet  MATH  Google Scholar 

  42. Akagi, G., Schimperna, G., Segatti, A.: Fractional Cahn–Hilliard, Allen–Cahn and porous medium equations. J. Differ. Equ. 261(6), 2935–2985 (2016)

    MathSciNet  MATH  Google Scholar 

  43. Zhao, J., Chen, L., Wang, H.: On power law scaling dynamics for time-fractional phase field models during coarsening. Commun. Nonlinear Sci. Numer. Simulat. 70, 257–270 (2019)

    MathSciNet  MATH  Google Scholar 

  44. Zhen, G., Lowengrub, J., Wang, C., Wise, S.: Second order convex splitting schemes for periodic nonlocal Cahn-Hilliard and Allen-Cahn equations. J. Comput. Phys. 277, 48–71 (2014)

    MathSciNet  MATH  Google Scholar 

  45. Zhang, H., Jiang, X.: A high-efficiency second-order numerical scheme for time-fractional phase field models by using extended SAV method. Nonlinear Dyn. 102, 589–603 (2020)

    Google Scholar 

  46. Liu, Z., Li, X., Huang, J.: Accurate and efficient algorithms with unconditional energy stability for the time fractional Cahn–Hilliard and Allen–Cahn equations. Numer. Methods Partial Differ. Equ. 37(3), 2613–2633 (2021)

    MathSciNet  Google Scholar 

  47. Antil, H., Baetrls, S.: Spectral approximation of fractional PDEs in image processing and phase field modeling. J. Comput. Methods Appl. Math. 17(4), 661–678 (2017)

    MathSciNet  MATH  Google Scholar 

  48. Liu, H., Cheng, A., Wang, H., Zhao, J.: Time-fractional Allen-Cahn and Cahn-Hilliard phase-field models and their numerical investigation. Comput. Math. Appl. 76(8), 1876–1892 (2018)

    MathSciNet  MATH  Google Scholar 

  49. Song, F., Xu, C., Karniadakis, G.: A fractional phase-field model for two-phase flows with tunable sharpness: Algorithms and simulations. Comput. Methods Appl. Mech. Engrg. 305, 376–404 (2016)

    MathSciNet  MATH  Google Scholar 

  50. Wu, B., Chen, Q., Wang, Z.: Uniqueness and stability of an inverse problem for a phase field model using data from one component. Comput. Math. Appl. 66, 2126–2138 (2013)

    MathSciNet  MATH  Google Scholar 

  51. Baranibalan, N., Sakthivel, K., Balachandran, K., Kim, J.: Reconstruction of two time independent coefficients in an inverse problem for a phase field system. Nonlinear Anal. 72, 2841–2851 (2010)

    MathSciNet  MATH  Google Scholar 

  52. Colombo, F.: Direct and inverse problems for a phase-field model with memory. J. Math. Anal. Appl. 260, 517–545 (2001)

    MathSciNet  MATH  Google Scholar 

  53. Khodadadian, A., Noii, N., Parvizi, M., Abbaszadeh, M., Wick, T., Heitzinger, C.: A Bayesian estimation method for variational phase-field fracture problems. Comput. Mech. 66(4), 827–849 (2020)

    MathSciNet  MATH  Google Scholar 

  54. Bandai, T., Ghezzehei, T.: Physics-informed neural networks with monotonicity constraints for Richardson–Richards equation: estimation of constitutive relationships and soil water flux density from volumetric water content measurements. Water Resour. Res. 57(2), 20 (2021)

    Google Scholar 

  55. Ruder, S.: An overview of gradient descent optimization algorithms. arXiv:1609.04747v2 (2016)

  56. Huisman, M., van Rijn, J., Plaat, A.: A survey of deep meta-learning. Artif. Intell. Rev. 54(6), 4483–4541 (2021)

    Google Scholar 

Download references

Funding

This study was funded by the National Natural Science Foundation of China (grant number 12120101001, 12001326), Natural Science Foundation of Shandong Province (grant number ZR2020QA032), and China Postdoctoral Science Foundation (grant number BX20190191, 2020M672038).

Author information

Authors and Affiliations

  1. School of Mathematics, Shandong University, Jinan, 250100, People’s Republic of China

    Shupeng Wang, Hui Zhang & Xiaoyun Jiang

Authors
  1. Shupeng Wang
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Hui Zhang
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Xiaoyun Jiang
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Hui Zhang.

Ethics declarations

Conflict of Interest

The authors declare that they have no conflict of interests regarding the publication of this paper.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Springer Nature or its licensor holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Wang, S., Zhang, H. & Jiang, X. Fractional physics-informed neural networks for time-fractional phase field models. Nonlinear Dyn 110, 2715–2739 (2022). https://doi.org/10.1007/s11071-022-07746-3

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11071-022-07746-3

Keywords

Search

Navigation