Skip to main content
SpringerLink
Log in

Approximate solution of KdV-Burgers equation using improved PINNs algorithm

  • Original Research
  • Published:
Indian Journal of Pure and Applied Mathematics Aims and scope Submit manuscript

Abstract

Finding solutions to partial differential equations (PDEs) has long been a challenging endeavor. Despite various proposed methods, there isn’t a universal approach capable of solving all types of PDEs. Recently, deep learning methods have emerged as a powerful tool for the solution of PDEs. Among them, the physics-informed neural networks (PINNs) stand out, integrating fundamental physical laws into neural networks to enforce equation dynamics using space-time data. This paper focuses on utilizing an enhanced version of the PINNs algorithm to approximate solutions for the nonlinear KdV-Burgers equation, a well-known nonlinear PDE with applications ranging from explaining wave propagation in elastic tubes filled with fluid to modeling waves in shallow viscous fluids. Our experiments yield promising results, showcased in the numerical section, where we systematically compare the performance of the PINNs method against our improved variant for each problem instance.

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

Similar content being viewed by others

Data Availibility Statement

No Data is associated with the manuscript.

References

  1. Z. Leini, S. Xiaolei, Study on speech recognition method of artificial intelligence deep learning, in: Journal of Physics: Conference Series, Vol. 1754, IOP Publishing, 2021, p. 012183.

  2. S. Grigorescu, B. Trasnea, T. Cocias, G. Macesanu, A survey of deep learning techniques for autonomous driving, Journal of Field Robotics 37 (3) (2020) 362–386.

    Article  Google Scholar 

  3. M. N. Y. Ali, M. L. Rahman, J. Chaki, N. Dey, K. Santosh, Machine translation using deep learning for universal networking language based on their structure, International Journal of Machine Learning and Cybernetics 12 (8) (2021) 2365–2376.

    Article  Google Scholar 

  4. B. Alipanahi, A. Delong, M. T. Weirauch, B. J. Frey, Predicting the sequence specificities of dna-and rna-binding proteins by deep learning, Nature biotechnology 33 (8) (2015) 831–838.

    Article  Google Scholar 

  5. S. Meng, N. Zhang, Y. Ren, X-densenet: deep learning for garbage classification based on visual images, in: Journal of Physics: Conference Series, Vol. 1575, IOP Publishing, 2020, p. 012139.

  6. A. Krizhevsky, I. Sutskever, G. E. Hinton, Imagenet classification with deep convolutional neural networks, Advances in neural information processing systems 25 (2012).

  7. L. Cai, C. Liu, R. Yuan, H. Ding, Human action recognition using lie group features and convolutional neural networks, Nonlinear Dynamics 99 (2020) 3253–3263.

    Article  Google Scholar 

  8. M. Raissi, P. Perdikaris, G. E. Karniadakis, Machine learning of linear differential equations using gaussian processes, Journal of Computational Physics 348 (2017) 683–693.

    Article  MathSciNet  Google Scholar 

  9. M. Raissi, P. Perdikaris, G. E. Karniadakis, Inferring solutions of differential equations using noisy multi-fidelity data, Journal of Computational Physics 335 (2017) 736–746.

    Article  MathSciNet  Google Scholar 

  10. I. Wasim, M. Abbas, M. Amin, Hybrid b-spline collocation method for solving the generalized burgers-fisher and burgers-huxley equations, Mathematical Problems in Engineering 2018 (2018) 1–18.

    Article  MathSciNet  Google Scholar 

  11. M. R. Ali, A. R. Hadhoud, W.-X. Ma, Evolutionary numerical approach for solving nonlinear singular periodic boundary value problems, Journal of Intelligent & Fuzzy Systems 39 (5) (2020) 7723–7731.

    Article  Google Scholar 

  12. M. R. Ali, R. Sadat, Lie symmetry analysis, new group invariant for the (3+ 1)-dimensional and variable coefficients for liquids with gas bubbles models, Chinese Journal of Physics 71 (2021) 539–547.

    Article  MathSciNet  Google Scholar 

  13. M. R. Ali, W.-X. Ma, Detection of new multi-wave solutions in an unbounded domain, Modern Physics Letters B 33 (34) (2019) 1950425.

    Article  MathSciNet  Google Scholar 

  14. R. Guo, Z. Lin, T. Shan, M. Li, F. Yang, S. Xu, A. Abubakar, Solving combined field integral equation with deep neural network for 2-d conducting object, IEEE Antennas and Wireless Propagation Letters 20 (4) (2021) 538–542.

    Article  Google Scholar 

  15. J. Jin, L. Zhao, M. Li, F. Yu, Z. Xi, Improved zeroing neural networks for finite time solving nonlinear equations, Neural Computing and Applications 32 (2020) 4151–4160.

    Article  Google Scholar 

  16. J. Han, A. Jentzen, W. E, Solving high-dimensional partial differential equations using deep learning, Proceedings of the National Academy of Sciences 115 (34) (2018) 8505–8510.

  17. Y. Fang, G.-Z. Wu, Y.-Y. Wang, C.-Q. Dai, Data-driven femtosecond optical soliton excitations and parameters discovery of the high-order nlse using the pinn, Nonlinear Dynamics 105 (1) (2021) 603–616.

    Article  Google Scholar 

  18. J. Sirignano, K. Spiliopoulos, Dgm: A deep learning algorithm for solving partial differential equations, Journal of computational physics 375 (2018) 1339–1364.

    Article  MathSciNet  Google Scholar 

  19. M. A. Nabian, H. Meidani, A deep learning solution approach for high-dimensional random differential equations, Probabilistic Engineering Mechanics 57 (2019) 14–25.

    Article  Google Scholar 

  20. W. Cai, Z.-Q. J. Xu, Multi-scale deep neural networks for solving high dimensional pdes, arXiv preprint arXiv:1910.11710 (2019).

  21. K. Li, K. Tang, T. Wu, Q. Liao, D3m: A deep domain decomposition method for partial differential equations, IEEE Access 8 (2019) 5283–5294.

    Article  Google Scholar 

  22. E. Weinan, B. Yu, The deep ritz method: a deep learning-based numerical algorithm for solving variational problems, Communications in Mathematics and Statistics 6 (1) (2018) 1–12.

    Article  MathSciNet  Google Scholar 

  23. H. Zhang, Y. Xu, Q. Liu, Y. Li, Deep learning framework for solving fokker–planck equations with low-rank separation representation, Engineering Applications of Artificial Intelligence 121 (2023) 106036.

  24. Y. Zang, G. Bao, X. Ye, H. Zhou, Weak adversarial networks for high-dimensional partial differential equations, Journal of Computational Physics 411 (2020) 109409.

    Article  MathSciNet  Google Scholar 

  25. 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 378 (2019) 686–707.

    Article  MathSciNet  Google Scholar 

  26. Z. Mao, A. D. Jagtap, G. E. Karniadakis, Physics-informed neural networks for high-speed flows, Computer Methods in Applied Mechanics and Engineering 360 (2020) 112789.

    Article  MathSciNet  Google Scholar 

  27. X. Meng, Z. Li, D. Zhang, G. E. Karniadakis, Ppinn: Parareal physics-informed neural network for time-dependent pdes, Computer Methods in Applied Mechanics and Engineering 370 (2020) 113250.

    Article  MathSciNet  Google Scholar 

  28. J. Li, Y. Chen, A physics-constrained deep residual network for solving the sine-gordon equation, Communications in Theoretical Physics 73 (1) (2020) 015001.

    Article  MathSciNet  Google Scholar 

  29. J. Li, Y. Chen, Solving second-order nonlinear evolution partial differential equations using deep learning, Communications in Theoretical Physics 72 (10) (2020) 105005.

    Article  MathSciNet  Google Scholar 

  30. C. H. Su, C. S. Gardner, Korteweg-de vries equation and generalizations. iii. derivation of the korteweg-de vries equation and burgers equation, Journal of Mathematical Physics 10 (3) (1969) 536–539.

  31. R. Johnson, A non-linear equation incorporating damping and dispersion, Journal of Fluid Mechanics 42 (1) (1970) 49–60.

    Article  MathSciNet  Google Scholar 

  32. R. Johnson, Shallow water waves on a viscous fluid-the undular bore, The Physics of Fluids 15 (10) (1972) 1693–1699.

    Article  Google Scholar 

  33. H. Grad, P. N. Hu, Unified shock profile in a plasma, The Physics of Fluids 10 (12) (1967) 2596–2602.

    Article  Google Scholar 

  34. A. El-Ajou, O. A. Arqub, S. Momani, Approximate analytical solution of the nonlinear fractional kdv–burgers equation: a new iterative algorithm, Journal of Computational Physics 293 (2015) 81–95.

  35. W. Hereman, A. Nuseir, Symbolic methods to construct exact solutions of nonlinear partial differential equations, Mathematics and Computers in Simulation 43 (1) (1997) 13–27.

    Article  MathSciNet  Google Scholar 

  36. T. Kawahara, Oscillatory solitary waves in dispersive media, Journal of the physical society of Japan 33 (1) (1972) 260–264.

    Article  Google Scholar 

  37. J. M. Burgers, A mathematical model illustrating the theory of turbulence, Advances in applied mechanics 1 (1948) 171–199.

    Article  MathSciNet  Google Scholar 

  38. J. D. Cole, On a quasi-linear parabolic equation occurring in aerodynamics, Quarterly of applied mathematics 9 (3) (1951) 225–236.

    Article  MathSciNet  Google Scholar 

  39. H. Ahmad, A. R. Seadawy, T. A. Khan, Numerical solution of korteweg–de vries-burgers equation by the modified variational iteration algorithm-ii arising in shallow water waves, Physica Scripta 95 (4) (2020) 045210.

  40. S. Ö. Korkut, N. İmamoğlu Karabaş, A reliable explicit method to approximate the general type of the kdv–burgers’ equation, Iranian Journal of Science and Technology, Transactions A: Science (2022) 1–11.

  41. F. Benkhaldoun, M. Seaid, New finite-volume relaxation methods for the third-order differential equations, Commun. Comput. Phys 4 (820-837) (2008) 3.

    MathSciNet  Google Scholar 

  42. M. Bektas, M. Inc, Y. Cherruault, Geometrical interpretation and approximate solution of non-linear kdv equation, Kybernetes 34 (7/8) (2005) 941–950.

    Article  Google Scholar 

  43. S. Abbasbandy, The application of homotopy analysis method to solve a generalized hirota–satsuma coupled kdv equation, Physics Letters A 361 (6) (2007) 478–483.

  44. W. Schiesser, Method of lines solution of the korteweg-de vries equation, Computers & Mathematics with Applications 28 (10-12) (1994) 147–154.

    Article  MathSciNet  Google Scholar 

  45. J. Biazar, H. Ghazvini, Exact solutions for nonlinear burgers’ equation by homotopy perturbation method, Numerical methods for partial differential equations 25 (4) (2009) 833–842.

    Article  MathSciNet  Google Scholar 

  46. R. Jiwari, R. Mittal, K. K. Sharma, A numerical scheme based on weighted average differential quadrature method for the numerical solution of burgers’ equation, Applied Mathematics and Computation 219 (12) (2013) 6680–6691.

    Article  MathSciNet  Google Scholar 

  47. M. Z. Gorgulu, I. Dag, D. Irk, Wave propagation by way of exponential b-spline galerkin method, in: Journal of Physics: Conference Series, Vol. 766, IOP Publishing, 2016, p. 012031.

  48. A. D. Jagtap, K. Kawaguchi, G. E. Karniadakis, Adaptive activation functions accelerate convergence in deep and physics-informed neural networks, Journal of Computational Physics 404 (2020) 109136.

    Article  MathSciNet  Google Scholar 

  49. T. A. Driscoll, N. Hale, L. N. Trefethen, Chebfun guide (2014).

  50. M. Stein, Large sample properties of simulations using latin hypercube sampling, Technometrics 29 (2) (1987) 143–151.

    Article  MathSciNet  Google Scholar 

  51. X. Glorot, Y. Bengio, Understanding the difficulty of training deep feedforward neural networks, in: Proceedings of the thirteenth international conference on artificial intelligence and statistics, JMLR Workshop and Conference Proceedings, 2010, pp. 249–256.

  52. D. C. Liu, J. Nocedal, On the limited memory bfgs method for large scale optimization, Mathematical programming 45 (1-3) (1989) 503–528.

    Article  MathSciNet  Google Scholar 

Download references

Funding

The authors declare that they have no funding.

Author information

Authors and Affiliations

  1. Department of Mathematics & Scientific Computing, National Institute of Technology Hamirpur, Hamirpur, H.P., 177001, India

    Harender Kumar

  2. Department of Mathematics, Dr. B. R. Ambedkar National Institute of Technology, Jalandhar, Punjab, 144011, India

    Neha Yadav

Authors
  1. Harender Kumar
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Neha Yadav
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Neha Yadav.

Ethics declarations

Conflict of Interest

The authors declare that they have no conflict of interest.

Additional information

Communicated by K Sandeep.

Rights and permissions

Springer Nature or its licensor (e.g. a society or other partner) 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

Kumar, H., Yadav, N. Approximate solution of KdV-Burgers equation using improved PINNs algorithm. Indian J Pure Appl Math (2024). https://doi.org/10.1007/s13226-024-00541-3

Download citation

  • Received:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s13226-024-00541-3

Keywords

Search

Navigation