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A machine learning framework for efficiently solving Fokker–Planck equations

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Abstract

This paper addresses the challenge of solving Fokker–Planck equations, which are prevalent mathematical models across a myriad of scientific fields. Due to factors like fractional-order derivatives and non-linearities, obtaining exact solutions to this problem can be complex. To overcome these challenges, our framework first discretizes the given equation using the Crank-Nicolson finite difference method, transforming it into a system of ordinary differential equations. Here, the approximation of time dynamics is done using forward difference or an L1 discretization technique for integer or fractional-order derivatives, respectively. Subsequently, these ordinary differential equations are solved using a novel strategy based on a kernel-based machine learning algorithm, named collocation least-squares support vector regression. The effectiveness of the proposed approach is demonstrated through multiple numerical experiments, highlighting its accuracy and efficiency. This performance establishes its potential as a valuable tool for tackling Fokker–Planck equations in diverse applications.

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Notes

  1. https://github.com/alirezaafzalaghaei/fokker-planck.

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Authors and Affiliations

  1. Department of Computer and Data Sciences, Shahid Beheshti University, G.C., Tehran, Iran

    Ali Nosrati Firoozsalari, Alireza Afzal Aghaei & Kourosh Parand

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  1. Ali Nosrati Firoozsalari
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  2. Alireza Afzal Aghaei
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  3. Kourosh Parand
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ANF: writing—original draft, resources, software AAA: formal analysis, software, writing—review & editing KP: supervision.

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Correspondence to Kourosh Parand.

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Firoozsalari, A.N., Aghaei, A.A. & Parand, K. A machine learning framework for efficiently solving Fokker–Planck equations. Comp. Appl. Math. 43, 389 (2024). https://doi.org/10.1007/s40314-024-02899-w

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