Operator learning for solving Fokker-Planck equations with various initial conditions

• URL: http://arxiv.org/abs/2606.09434v1
• Date: Mon, 08 Jun 2026 12:44:13 GMT
• Title: Operator learning for solving Fokker-Planck equations with various initial conditions
• Abstract summary: The Fokker-Planck equation (FPE) plays a pivotal role in describing the time evolution of density functions (PDFs)<n>We propose a conditional normalizing flow-based physics-informed neural network (PINN) framework for efficiently approximating the solution operator of the FPE.<n>A variety of numerical experiments are presented to illustrate the effectiveness and robustness of the proposed method.
• Score: 9.218744685480125
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: The Fokker-Planck equation (FPE) plays a pivotal role in describing the time evolution of probability density functions (PDFs) for systems governed by stochastic dynamics. In this work, we propose a conditional normalizing flow-based physics-informed neural network (PINN) framework for efficiently approximating the solution operator of the FPE for a whole range of initial conditions. Leveraging the Chapman-Kolmogorov equation for Markovian stochastic processes, the problem is reformulated into approximating a transition PDF starting at initial time from a Dirac mass centered at an arbitrary point. The PDF of an associated linearized stochastic differential equation (SDE) is employed as the base distribution for the normalizing flow, providing a good approximation of the target PDF, especially for small times, and thereby avoiding the singularity of the map associated with the Dirac delta initial distribution. Furthermore, a time-weighted loss function is introduced to mitigate numerical instabilities arising at small times, achieving a balance between causality and training difficulty as time progresses. A variety of numerical experiments are presented to illustrate the effectiveness and robustness of the proposed method.

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