Numerical simulation by Deep Learning of a time periodic p(x)-Laplace equation

The objective of this paper is to focus on the study of a periodic temporal parabolic equation involving a variable exponent $p(x)$.  After proving the existence and uniqueness of the solution, we provide a method for its numerical simulation using emerging deep learning technologies.

periodic solution
p(x)-Laplace operator
глибинне навчання
  1. Brunner H., Makroglou A., Miller R. K.  On mixed collocation methods for Volterra integral equations with periodic solution.  Applied Numerical Mathematics.  24 (2–3), 115–130 (1997).
  2. Dababneh A., Zraiqat A., Farah A., Al-Zoubi H., Abu Hammad M. M.  Numerical methods for finding periodic solutions of ordinary differential equations with strong nonlinearity.  Journal of Mathematical and Computational Science.  11 (6), 6910–6922 (2021).
  3. Samoilenko A. M.  Certain questions of the theory of periodic and quasi-periodic systems.  D.Sc. Dissertation, Kiev (1967).
  4. El Ghabi M., Alaa H., Alaa N. E.  Semilinear periodic equation with arbitrary nonlinear growth and data measure: mathematical analysis and numerical simulation.  Mathematical Modeling and Computing.  10 (3), 956–964 (2023).
  5. Aggarwal C. C.  Neural Networks and Deep Learning.  A Textbook.  Springer, Cham (2018).
  6. Nascimento R. G., Fricke K., Viana F. A. C.  A tutorial on solving ordinary differential equations using Python and hybrid physics-informed neural network.  Engineering Applications of Artificial Intelligence.  96, 103996 (2020).
  7. Ranade R., Hill C., He H., Maleki A., Chang N., Pathak J.  A composable autoencoder-based iterative algorithm for accelerating numerical simulations.  Preprint arXiv:2110.03780 (2021).
  8. Li S., Song W., Fang L., Chen Y., Ghamisi P., Benediktsson J. A.  Deep learning for hyperspectral image classification: An overview.  IEEE Transactions on Geoscience and Remote Sensing.  57 (9), 6690–6709 (2019).
  9. Goldberg Y.  A primer on neural network models for natural language processing.  Journal of Artificial Intelligence Research.  57, 345–420 (2016).
  10. Helbing G., Ritter M.  Deep Learning for fault detection in wind turbines.  Renewable and Sustainable Energy Reviews.  98, 189–198 (2018).
  11. Hornik K., Stinchcombe M., White H.  Universal approximation of an unknown mapping and its derivatives using multilayer feedforward networks.  Neural Networks.  3 (5), 551–560 (1990).
  12. Pham B., Nguyen T., Nguyen T. T., Nguyen B. T.  Solve systems of ordinary differential equations using deep neural networks.  2020 7th NAFOSTED Conference on Information and Computer Science (NICS).  42–47 (2020).
  13. Dufera T. T.  Deep neural network for system of ordinary differential equations: Vectorized algorithm and simulation.  Machine Learning with Applications.  5, 100058 (2021).
  14. Nam H., Baek K. R., Bu S.  Error estimation using neural network technique for solving ordinary differential equations.  Advances in Continuous and Discrete Models.  2022, 45 (2022).
  15. Raissi M., Perdikaris P., Karniadakis G. E.  Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations.  Journal of Computational Physics.  378, 686–707 (2019).
  16. Raissi M., Perdikaris P., Karniadakis G. E.  Physics Informed Deep Learning (Part I): Data-driven Solutions of Nonlinear Partial Differential Equations.  Preprint arXiv:1711.10561 (2017).
  17. Hariri I., Radid A., Rhofir K.  Physics-informed neural networks for the reaction-diffusion Brusselator model.  Mathematical Modeling and Computing.  11 (2), 448–454 (2024).
  18. Hariri I., Radid A., Rhofir K.  A physical laws into Deep Neural Networks for solving generalized Burgers–Huxley equation.  Mathematical Modeling and Computing.  11 (2), 505–511 (2024).
  19. Jagtap A. D., Kawaguchi K., Karniadakis G. E.  Adaptive activation functions accelerate convergence in deep and physics-informed neural networks.  Journal of Computational Physics.  404, 109136 (2020).
  20. Radulescu V., Repovs D. D.  Partial Differential Equations with Variable Exponents: Variational Methods and Qualitative Analysis.  CRC Press Taylor and Francis Group (2015).
  21. Lu L., Meng X., Mao Z., Karniadakis G. E.  DeepXDE: A deep learning library for solving differential equations.  SIAM Review.  63 (1), 208–228 (2021).
  22. Bendahmane M., Wittbold P., Zimmermann A.  Renormalized solutions for a nonlinear parabolic equation with variable exponents and $L^{1}$-data.  Journal of Differential Equations.  249 (6), 1483–1515 (2010).
  23. Lions J. L.  Quelques méthodes de résolution de problèmes aux limites non linéaires.  Dunod, Paris (1969).
  24. Cybenko G.  Approximation by superpositions of a sigmoidal function.  Mathematics of Control, Signals and Systems.  2 (4), 303–314 (1989).
  25. Hornik K.  Approximation capabilities of multilayer feedforward networks.  Neural Networks.  4 (2), 251–257 (1991).
  26. Ziyin L., Hartwig T., Ueda M.  Neural networks fail to learn periodic functions and how to fix it.  Advances in Neural Information Processing Systems.  33, 1583–1594 (2020).
  27. Dong S., Ni N.  A method for representing periodic functions and enforcing exactly periodic boundary conditions with deep neural networks.  Journal of Computational Physics.  435, 110242 (2021).
  28. Lu L., Pestourie R., Yao W., Wang Z., Verdugo F., Johnson S. G.  Physics-informed neural networks with hard constraints for inverse design.  SIAM Journal on Scientific Computing.  43 (6), B1105–B1132 (2021).
  29. Sacchetti A., Bachmann B., Löffel K., Künzi U.-M., Paoli B.  Neural Networks to Solve Partial Differential Equations: A Comparison With Finite Elements.  IEEE Access.  10, 32271–32279 (2022).
  30. Hinton G., Srivastava N., Swersky K.  Neural networks for machine learning lecture 6a overview of mini-batch gradient descent.  (2012).
  31. Byrd R. H., Lu P., Nocedal J., Zhu C.  A limited memory algorithm for bound constrained optimization.  SIAM Journal on Scientific Computing.  16 (5), 1190–1208 (1995).
  32. Liu D. C., Nocedal J.  On the limited memory BFGS method for large scale optimization.  Mathematical Programming.  45 (1), 503–528 (1989).