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Efficient Solution of Fokker–Planck Equations in Two Dimensions

Author

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  • 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
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    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.
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