IDEAS home Printed from https://ideas.repec.org/a/gam/jmathe/v13y2025i3p491-d1581522.html
   My bibliography  Save this article

Efficient Solution of Fokker–Planck Equations in Two Dimensions

Author

Listed:
  • Donald Michael McFarland

    (Sound and Vibration Laboratory, College of Mechanical Engineering, Zhejiang University of Technology, Hangzhou 310014, China
    Ningbo Institute of Digital Twin, Eastern Institute of Technology, Ningbo 315201, China)

  • Fei Ye

    (Sound and Vibration Laboratory, College of Mechanical Engineering, Zhejiang University of Technology, Hangzhou 310014, China)

  • Chao Zong

    (Sound and Vibration Laboratory, College of Mechanical Engineering, Zhejiang University of Technology, Hangzhou 310014, China)

  • Rui Zhu

    (Sound and Vibration Laboratory, College of Mechanical Engineering, Zhejiang University of Technology, Hangzhou 310014, China)

  • Tao Han

    (Ningbo Institute of Digital Twin, Eastern Institute of Technology, Ningbo 315201, China)

  • Hangyu Fu

    (Sound and Vibration Laboratory, College of Mechanical Engineering, Zhejiang University of Technology, Hangzhou 310014, China)

  • Lawrence A. Bergman

    (Department of Aerospace Engineering, University of Illinois at Urbana-Champaign, Urbana, IL 61801, USA)

  • Huancai Lu

    (Ningbo Institute of Digital Twin, Eastern Institute of Technology, Ningbo 315201, China)

Abstract

Finite element analysis (FEA) of the Fokker–Planck equation governing the nonstationary joint probability density function of the responses of a dynamical system produces a large set of ordinary differential equations, and computations become impractical for systems with as few as four states. Nonetheless, FEA remains of interest for small systems—for example, for the generation of baseline performance data and reference solutions for the evaluation of machine learning-based methods. We examine the effectiveness of two techniques which, while they are well established, have not to our knowledge been applied to this problem previously: reduction of the equations onto a smaller basis comprising selected eigenvectors of one of the coefficient matrices, and splitting of the other coefficient matrix. The reduction was only moderately effective, requiring a much larger basis than was expected and producing solutions with clear artifacts. Operator splitting, however, performed very well. While the methods can be combined, our results indicate that splitting alone is an effective and generally preferable approach.

Suggested Citation

  • Donald Michael McFarland & Fei Ye & Chao Zong & Rui Zhu & Tao Han & Hangyu Fu & Lawrence A. Bergman & Huancai Lu, 2025. "Efficient Solution of Fokker–Planck Equations in Two Dimensions," Mathematics, MDPI, vol. 13(3), pages 1-20, January.
    • Handle: RePEc:gam:jmathe:v:13:y:2025:i:3:p:491-:d:1581522
    as

    Download full text from publisher

    File URL: https://www.mdpi.com/2227-7390/13/3/491/pdf
    Download Restriction: no
    File URL: https://www.mdpi.com/2227-7390/13/3/491/
    Download Restriction: no
    ---><---

    References listed on IDEAS

    as
    1. Sepehrian, Behnam & Radpoor, Marzieh Karimi, 2015. "Numerical solution of non-linear Fokker–Planck equation using finite differences method and the cubic spline functions," Applied Mathematics and Computation, Elsevier, vol. 262(C), pages 187-190.
    2. Justin Sirignano & Konstantinos Spiliopoulos, 2017. "DGM: A deep learning algorithm for solving partial differential equations," Papers 1708.07469, arXiv.org, revised Sep 2018.
    Full references (including those not matched with items on IDEAS)

    Most related items

    These are the items that most often cite the same works as this one and are cited by the same works as this one.
    1. Li, Wei & Zhang, Ying & Huang, Dongmei & Rajic, Vesna, 2022. "Study on stationary probability density of a stochastic tumor-immune model with simulation by ANN algorithm," Chaos, Solitons & Fractals, Elsevier, vol. 159(C).
    2. Kristina O. F. Williams & Benjamin F. Akers, 2023. "Numerical Simulation of the Korteweg–de Vries Equation with Machine Learning," Mathematics, MDPI, vol. 11(13), pages 1-14, June.
    3. William Lefebvre & Enzo Miller, 2021. "Linear-quadratic stochastic delayed control and deep learning resolution," Working Papers hal-03145949, HAL.
    4. Zhouzhou Gu & Mathieu Lauri`ere & Sebastian Merkel & Jonathan Payne, 2024. "Global Solutions to Master Equations for Continuous Time Heterogeneous Agent Macroeconomic Models," Papers 2406.13726, arXiv.org.
    5. Parand, K. & Aghaei, A.A. & Jani, M. & Ghodsi, A., 2021. "A new approach to the numerical solution of Fredholm integral equations using least squares-support vector regression," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 180(C), pages 114-128.
    6. William Lefebvre & Enzo Miller, 2021. "Linear-quadratic stochastic delayed control and deep learning resolution," Papers 2102.09851, arXiv.org, revised Feb 2021.
    7. A. Max Reppen & H. Mete Soner & Valentin Tissot-Daguette, 2022. "Deep Stochastic Optimization in Finance," Papers 2205.04604, arXiv.org.
    8. Sebastian Jaimungal, 2022. "Reinforcement learning and stochastic optimisation," Finance and Stochastics, Springer, vol. 26(1), pages 103-129, January.
    9. Shuaiqiang Liu & Cornelis W. Oosterlee & Sander M. Bohte, 2019. "Pricing Options and Computing Implied Volatilities using Neural Networks," Risks, MDPI, vol. 7(1), pages 1-22, February.
    10. Bastien Baldacci & Joffrey Derchu & Iuliia Manziuk, 2020. "An approximate solution for options market-making in high dimension," Papers 2009.00907, arXiv.org.
    11. Alexandre Pannier & Cristopher Salvi, 2024. "A path-dependent PDE solver based on signature kernels," Papers 2403.11738, arXiv.org, revised Oct 2024.
    12. Rong Du & Duy-Minh Dang, 2023. "Fourier Neural Network Approximation of Transition Densities in Finance," Papers 2309.03966, arXiv.org, revised Sep 2024.
    13. Gero Junike & Hauke Stier, 2024. "Enhancing Fourier pricing with machine learning," Papers 2412.05070, arXiv.org.
    14. Ali Al-Aradi & Adolfo Correia & Danilo de Frietas Naiff & Gabriel Jardim & Yuri Saporito, 2019. "Extensions of the Deep Galerkin Method," Papers 1912.01455, arXiv.org, revised Apr 2022.
    15. Yuga Iguchi & Riu Naito & Yusuke Okano & Akihiko Takahashi & Toshihiro Yamada, 2021. "Deep Asymptotic Expansion: Application to Financial Mathematics," CIRJE F-Series CIRJE-F-1178, CIRJE, Faculty of Economics, University of Tokyo.
    16. Martin Hutzenthaler & Arnulf Jentzen & Thomas Kruse & Tuan Anh Nguyen, 2020. "A proof that rectified deep neural networks overcome the curse of dimensionality in the numerical approximation of semilinear heat equations," Partial Differential Equations and Applications, Springer, vol. 1(2), pages 1-34, April.
    17. Sebastian Becker & Patrick Cheridito & Arnulf Jentzen, 2020. "Pricing and Hedging American-Style Options with Deep Learning," JRFM, MDPI, vol. 13(7), pages 1-12, July.
    18. Salah A. Faroughi & Ramin Soltanmohammadi & Pingki Datta & Seyed Kourosh Mahjour & Shirko Faroughi, 2023. "Physics-Informed Neural Networks with Periodic Activation Functions for Solute Transport in Heterogeneous Porous Media," Mathematics, MDPI, vol. 12(1), pages 1-23, December.
    19. Jiequn Han & Ruimeng Hu & Jihao Long, 2020. "Convergence of Deep Fictitious Play for Stochastic Differential Games," Papers 2008.05519, arXiv.org, revised Mar 2021.
    20. Lukas Gonon, 2024. "Deep neural network expressivity for optimal stopping problems," Finance and Stochastics, Springer, vol. 28(3), pages 865-910, July.

    Corrections

    All material on this site has been provided by the respective publishers and authors. You can help correct errors and omissions. When requesting a correction, please mention this item's handle: RePEc:gam:jmathe:v:13:y:2025:i:3:p:491-:d:1581522. See general information about how to correct material in RePEc.

    If you have authored this item and are not yet registered with RePEc, we encourage you to do it here. This allows to link your profile to this item. It also allows you to accept potential citations to this item that we are uncertain about.

    If CitEc recognized a bibliographic reference but did not link an item in RePEc to it, you can help with this form .

    If you know of missing items citing this one, you can help us creating those links by adding the relevant references in the same way as above, for each refering item. If you are a registered author of this item, you may also want to check the "citations" tab in your RePEc Author Service profile, as there may be some citations waiting for confirmation.

    For technical questions regarding this item, or to correct its authors, title, abstract, bibliographic or download information, contact: MDPI Indexing Manager (email available below). General contact details of provider: https://www.mdpi.com .

    Please note that corrections may take a couple of weeks to filter through the various RePEc services.

    IDEAS is a RePEc service. RePEc uses bibliographic data supplied by the respective publishers.