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

2026-06-08 · Source: Machine Learning · Field: Technology & Digital — Artificial Intelligence & Machine Learning, Mathematics & Computational Sciences · Depth: Expert, quick

Summary

The paper introduces a conditional normalizing flow-based physics-informed neural network (PINN) framework designed to efficiently approximate the solution operator of the Fokker-Planck equation (FPE) across a wide range of initial conditions. This method reformulates the problem by leveraging the Chapman-Kolmogorov equation, focusing on approximating a transition probability density function (PDF) from a Dirac mass. It employs the PDF of an associated linearized stochastic differential equation (SDE) as the normalizing flow's base distribution, which effectively approximates the target PDF at small times and mitigates the singularity issue of Dirac delta initial distributions. Furthermore, a time-weighted loss function is incorporated to address numerical instabilities occurring at small times, balancing causality with training complexity. Numerical experiments demonstrate the framework's effectiveness and robustness.

Key takeaway

For research scientists developing solutions for stochastic dynamics, this conditional normalizing flow PINN offers a robust approach to solving Fokker-Planck equations across varied initial conditions. You should consider integrating its Chapman-Kolmogorov reformulation and linearized SDE base distribution to handle Dirac delta singularities and improve small-time accuracy. Implementing a time-weighted loss function can further stabilize training and enhance overall model performance.

Key insights

A conditional normalizing flow PINN efficiently solves Fokker-Planck equations for diverse initial conditions by leveraging SDEs and time-weighted loss.

Principles

• Chapman-Kolmogorov equation reformulates FPE for operator learning.
• Linearized SDE PDFs can serve as effective base distributions.
• Time-weighted loss mitigates small-time numerical instabilities.

Method

The framework uses a conditional normalizing flow PINN, reformulating the FPE problem via the Chapman-Kolmogorov equation to approximate a transition PDF. It employs a linearized SDE's PDF as a base distribution and a time-weighted loss function.

Topics

• Operator Learning
• Fokker-Planck Equation
• Normalizing Flows
• Physics-Informed Neural Networks
• Stochastic Differential Equations
• Probability Density Functions

Best for: AI Scientist, Research Scientist

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Editorial summary, takeaway, and curation by AIssential. Original article published by Machine Learning.