Numerical solution of the Fokker–Planck equation using physics-based mixture models

Armin Tabandeh, Neetesh Sharma, Leandro Iannacone, Paolo Gardoni

Research output: Contribution to journalArticlepeer-review


The Fokker–Planck equation governs the uncertainty propagation of dynamical systems driven by stochastic processes. The solution of the Fokker–Planck equation is a time-varying Probability Density Function (PDF) that is usually of high dimension with unbounded support. There are also properties that must be conserved in time for the joint and any marginal solution PDF (i.e., a time-varying PDF is a non-negative function that must integrate to unity at any given time.) Satisfying these properties poses a significant challenge to the numerical solution of the Fokker–Planck equation. Another challenge is to capture the tail behavior of the solution PDF with unbounded support, required to predict the probability of rare events like a system failure. Satisfying the conditions of a proper PDF with unbounded support limits the application of traditional grid-based numerical methods, like finite difference and finite element methods. This paper develops a novel numerical method based on physics-based mixture models for the transient and steady-state solutions of the Fokker–Planck equation. In the proposed numerical method, the trial function space consists of the convex combinations of some parametric PDFs (i.e., mixture models) that trivially satisfy the necessary conditions of the solution. Estimating the unknown parameters of the mixture model is via Bayesian inference while considering the constraints on model parameters. Bayesian inference facilitates integrating data on the responses of dynamical systems (e.g., from simulations) with the governing Fokker–Planck equation to estimate the unknown parameters. Since the solution PDF is not observable, combining it with observable data on system responses is far from trivial based on current numerical methods. The formulation of Bayesian inference also introduces a weighting function to reduce the estimation error in specific regions of interest like the tail of the solution PDF. To reduce the computational demand, the paper develops an importance sampling algorithm that generates a small set of collocation points at which the residual of the Fokker–Planck equation is evaluated. The performance of the proposed numerical method is demonstrated with several benchmark problems. The obtained results are compared with analytical solutions when available and otherwise with simulations.

Original languageEnglish (US)
Article number115424
JournalComputer Methods in Applied Mechanics and Engineering
StatePublished - Sep 1 2022


  • Fokker–Planck equation
  • Nonlinear systems
  • Physics-based mixture model
  • Random vibration
  • Reliability analysis

ASJC Scopus subject areas

  • Computational Mechanics
  • Mechanics of Materials
  • Mechanical Engineering
  • General Physics and Astronomy
  • Computer Science Applications


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