Skip to main content
Springer Nature Link
Log in

RETRACTED ARTICLE: Model Order Reduction Method Based on Machine Learning for Parameterized Time-Dependent Partial Differential Equations

  • Published:
Journal of Scientific Computing Aims and scope Submit manuscript

This article was retracted on 29 October 2022

This article has been updated

Abstract

In this paper, we propose a data-driven model order reduction method to solve parameterized time-dependent partial differential equations. We describe the system with the state variable equations, and represent a class of candidate models with the artificial neural network. The discrete \(L_2\) error between the output of artificial neural network and the high-fidelity solution is minimized with the state variable equations and initial conditions as constraints. Therefore, the model order reduction problem can be described as a kind of optimization problem with constraints, which can be solved by combining Levenberg–Marquardt algorithm and linear search algorithm, followed by sensitivity analysis of the artificial neural network parameters. Finally, by a number of calculating examples, compared to the model-based model order reduction method, data-driven model order reduction method is non-intrusive, is not limited to state variable degrees of freedom. We can find that the data-driven model order reduction method is better than the model-based model order reduction method in both computation time and precision, and has good approximation properties.

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

Access this article

Log in via an institution

Subscribe and save

Springer+ Basic
$34.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or eBook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

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

Similar content being viewed by others

Data Availability

The data that support the findings of this study are available from the corresponding author upon reasonable request.

Change history

References

  1. Quarteroni, A., Manzoni, A., Negri, F.: Reduced basis methods for partial differential equations: an introduction. Springer, Berlin (2015)

    MATH  Google Scholar 

  2. Kunisch, K., Volkwein, S.: Galerkin proper orthogonal decomposition methods for a general equation in fluid dynamics. Soc. Ind. Appl. Math. 40(2), 492–515 (2002)

    MathSciNet  MATH  Google Scholar 

  3. Abbasi, F., Velni, J.M.: Nonlinear model order reduction of Burgers’ equation using proper orthogonal decomposition, American Control Conferen, pp. 583–588 (2015)

  4. Berg, J., Nyström, K.: A unified deep artificial neural network approach to partial differential equations in complex geometries. Neurocomputing 317, 28–41 (2018)

    Article  Google Scholar 

  5. Rudy, S.H., Brunton, S.L., Proctor, J.L., Kutz, J.N.: Data-driven discovery of partial differential equations. Sci. Adv. 3(4), 1602614 (2017)

    Article  Google Scholar 

  6. Raissi, M., Karniadakis, G.E.: Hidden physics models: Machine learning of nonlinear partial differential equations. J. Comput. Phys. 357, 125–141 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  7. Raissi, M., Perdikaris, P., Karniadakis, G.E.: Machine learning of linear differential equations using Gaussian processes. J. Comput. Phys. 348, 683–693 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  8. Santo, N., Deparis, S., Pegolotti, L.: Data driven approximation of parametrized PDEs by reduced basis and neural networks. J. Comput. Phys. 416, 109550 (2020)

    Article  MathSciNet  MATH  Google Scholar 

  9. Guo, M., Hesthaven, J.S.: Data-driven reduced order modeling for time-dependent problems. Comput. Methods Appl. Mech. Eng. 345, 75–99 (2019)

    Article  MathSciNet  MATH  Google Scholar 

  10. Wu, P., Sun, J., Chang, X., Zhang, W., Arcucci, R., Guo, Y., Pain, C.C.: Data-driven reduced order model with temporal convolutional neural network. Comput. Methods Appl. Mech. Eng. 360, 112766 (2020)

    Article  MathSciNet  MATH  Google Scholar 

  11. Audouze, C., Vuyst, F.D., Nair, P.B.: Nonintrusive reduced-order modeling of parametrized time-dependent partial differential equations. Numer. Methods Partial Differ. Equ. 29(5), 1587–1628 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  12. Raissi, M., Perdikaris, P., Karniadakis, G.E.: Multistep neural networks for data-driven discovery of nonlinear dynamical systems, (2018), arXiv:1801.01236v1

  13. San, O., Maulik, R.: Machine learning closures for model order reduction of thermal fluids. Appl. Math. Model. 60, 681–710 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  14. Wang, Q., Hesthaven, J.S., Ray, D.: Non-intrusive reduced order modeling of unsteady flows using artificial neural networks with application to acombustion problem. J. Comput. Phys. 384, 289–307 (2019)

    Article  MathSciNet  MATH  Google Scholar 

  15. Hesthaven, J.S., Ubbiali, S.: Non-intrusive reduced order modeling of nonlinear problems using neural networks. Article in Journal of Computational. Phys. 363, 55–78 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  16. Frescaa, S., Dedè, L., Manzonia, A.: A comprehensive deep learning-based approach to reduced order modeling of nonlinear time-dependent parametrized PDEs. J. Sci. Comput. 87, 61 (2021)

    Article  MathSciNet  Google Scholar 

  17. Frescaa, S., Manzonia, A.: POD-DL-ROM: enhancing deep learning-based reduced order models for nonlinear parametrized PDEs by proper orthogonal decomposition, (2021), arXiv:2101.11845v1

  18. Fresca, S., Manzoni, A., Dedè, L., Quarteroni, A.: Deep learning-based reduced order models in cardiac electrophysiology. PLoS ONE 15(10), 0239416 (2020)

    Article  Google Scholar 

  19. 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. J. Comput. Phys. 378, 686–70 (2019)

    Article  MathSciNet  MATH  Google Scholar 

  20. Chen, W., Wang, Q., Hesthaven, J.S., Zhang, C.: Physics-informed machine learning for reduced-order modeling of nonlinear problems. J. Comput. Phys. 446, 110666 (2021)

    Article  MathSciNet  MATH  Google Scholar 

  21. Mücke, N.T., Bohté, S.M., Oosterlee, C.W.: Reduced order modeling for parameterized time-dependent PDEs using spatially and memory aware deep learning. J. Comput. Sci. 53, 101408 (2021)

    Article  Google Scholar 

  22. Benner, P., Gugercin, S., Willcox, K.: A survey of projection-based model reduction methods for parametric dynamical systems. Soc. Ind. Appl. Math. 57(4), 483–531 (2015)

    MathSciNet  MATH  Google Scholar 

  23. Peherstorfer, B., Gugercin, S., Willcox, K.: Data-driven reduced model construction with time-domain Loewner models. SIAM J. Sci. Comput. 39(5), A2152–A2178 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  24. Regazzoni, F., Dedè, L., Quarteroni, A.: Machine learning for fast and reliable solution of time-dependent differential equations. J. Comput. Phys. 397, 108852 (2019)

    Article  MathSciNet  MATH  Google Scholar 

  25. Regazzoni, F., Dedè, L., Quarteroni, A.: Machine learning of multiscale active force generation models for the efficient simulation of cardiac electromechanics. Comput. Methods Appl. Mech. Eng. 370, 113268 (2020)

    Article  MathSciNet  MATH  Google Scholar 

  26. Nelles, O.: Nonlinear system identification: from classical approaches to neural networks, fuzzy models, and Gaussian processes. Springer, Berlin (2021)

    MATH  Google Scholar 

  27. Gosea, I.V.: Model order reduction of linear and nonlinear systems in the Loewner framework, (2018)http://urn-resolving.de/urn:nbn:de:gbv:579-opus-1007899

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

    Article  MathSciNet  MATH  Google Scholar 

  29. Cybenko, G.: Approximation by superpositions of a sigmoidal function. Math. Control Signals Systems 2(4), 303–314 (1989)

    Article  MathSciNet  MATH  Google Scholar 

  30. Barron, A.R.: Universal approximation bounds for superpositions of a sigmoidal function. IEEE Trans. Inf. Theory 39(3), 930–945 (1993)

    Article  MathSciNet  MATH  Google Scholar 

  31. Hagan, M.T., Demuth, H.B.: Neural network design. China Machine Press, Beijing, China (2002)

    Google Scholar 

  32. Goodfellow, I., Bengio, Y., Courville, A.: Deep learning, the MIT Press, (2016), http://www.deeplearningbook.org

  33. Bottou, L., Curtis, F.E., Nocedal, J.: Optimization methods for large-scale machine learning. Soc. Ind. Appl. Math. 60(2), 223–311 (2018)

    MathSciNet  MATH  Google Scholar 

  34. Zhang, J., Yan, G.: Lattice Boltzmann method for one and two-dimensional Burgers’ equation. Phys. A 387, 4771–4786 (2008)

  35. Manzoni, A., Pagani, S., Lassila, T.: Accurate solution of Bayesian inverse uncertainty quantification problems combining reduced basis methods. SIAM/ASA J. Uncertain. Quant. 4(1), 380–412 (2016)

    Article  MATH  Google Scholar 

  36. Tiumentsev, Y.V., Egorchev, M.V.: Neural network modeling and identification of dynamical systems. Academic Press, London, UK (2019)

    Google Scholar 

Download references

Acknowledgements

This work is supported by the Research Fund from Key Laboratory of Xinjiang Province (No. 2020D04002), the Natural Science foundation of China (No. U19A2079, 11671345, 91630205, 91852106, 92152301).

Funding

The authors have not disclosed any funding.

Author information

Authors and Affiliations

  1. College of Mathematics and System Sciences, Xinjiang University, Urumqi, 830046, People’s Republic of China

    Fangxiong Cheng & Xinlong Feng

  2. School of Aeronautics and Astronautics, Shanghai Jiao Tong University, Shanghai, 200240, People’s Republic of China

    Hui Xu

Authors
  1. Fangxiong Cheng
    View author publications

    You can also search for this author inPubMed Google Scholar

  2. Hui Xu
    View author publications

    You can also search for this author inPubMed Google Scholar

  3. Xinlong Feng
    View author publications

    You can also search for this author inPubMed Google Scholar

Corresponding author

Correspondence to Hui Xu.

Ethics declarations

Competing Interests

The authors have no conflicts to disclose.

Additional information

Publisher's Note

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

This article has been retracted. Please see the retraction notice for more detail:https://doi.org/10.1007/s10915-022-02036-x

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.

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Cheng, F., Xu, H. & Feng, X. RETRACTED ARTICLE: Model Order Reduction Method Based on Machine Learning for Parameterized Time-Dependent Partial Differential Equations. J Sci Comput 92, 64 (2022). https://doi.org/10.1007/s10915-022-01899-4

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s10915-022-01899-4

Keywords