Operator learning for solving Fokker-Planck equations with various initial conditions
2026-06-08 • Machine Learning
Machine Learning
AI summaryⓘ
The authors developed a new method using neural networks to better predict how probabilities change over time in random systems, described by the Fokker-Planck equation. They use a special technique called normalizing flows combined with physics knowledge to model many starting points at once. By basing their approach on a simpler related equation, they avoid tricky problems caused by starting conditions that are too sharp. They also designed a way to balance early-time accuracy and training challenges. Their tests show this method works reliably for these types of probability problems.
Fokker-Planck equationProbability density functionStochastic dynamicsNormalizing flowPhysics-informed neural networkChapman-Kolmogorov equationStochastic differential equationDirac deltaTransition probabilityMachine learning
Authors
Li Zeng, Xiaoliang Wan, Yaobin Wang, Fabio Nobile, Tao Zhou
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.