# A transition-density-based operator learning method for Fokker-Planck equations with various initial conditions

arXiv: https://arxiv.org/abs/2606.09434 · HTML: https://mlsanity.com/paper/2606.09434/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 28 of its 48 references that are in the map.
- First posted on arXiv 8 Jun 2026.

## Builds on (most cited first, 28 of 28)

- [Fourier Neural Operator for Parametric Partial Differential Equations](https://mlsanity.com/paper/2010.08895/) (2020)
- [Density estimation using Real NVP](https://mlsanity.com/paper/1605.08803/) (2016)
- [DeepONet: Learning nonlinear operators for identifying differential equations based on the universal approximation theorem of operators](https://mlsanity.com/paper/1910.03193/) (2019)
- [DGM: A deep learning algorithm for solving partial differential equations](https://mlsanity.com/paper/1708.07469/) (2017)
- [On the eigenvector bias of Fourier feature networks: From regression to solving multi-scale PDEs with physics-informed neural networks](https://mlsanity.com/paper/2012.10047/) (2020)
- [Mitigating Propagation Failures in Physics-informed Neural Networks using Retain-Resample-Release (R3) Sampling](https://mlsanity.com/paper/2207.02338/) (2022)
- [Neural Galerkin Schemes with Active Learning for High-Dimensional Evolution Equations](https://mlsanity.com/paper/2203.01360/) (2022)
- [Solving Inverse Stochastic Problems from Discrete Particle Observations Using the Fokker-Planck Equation and Physics-informed Neural Networks](https://mlsanity.com/paper/2008.10653/) (2020)
- [Probability flow solution of the Fokker-Planck equation](https://mlsanity.com/paper/2206.04642/) (2022)
- [Conditional Sampling with Monotone GANs: from Generative Models to Likelihood-Free Inference](https://mlsanity.com/paper/2006.06755/) (2020)
- [Structured Output Learning with Conditional Generative Flows](https://mlsanity.com/paper/1905.13288/) (2019)
- [Adaptive deep density approximation for Fokker-Planck equations](https://mlsanity.com/paper/2103.11181/) (2021)
- [Learning Green's functions associated with time-dependent partial differential equations](https://mlsanity.com/paper/2204.12789/) (2022)
- [Score-Based Physics-Informed Neural Networks for High-Dimensional Fokker-Planck Equations](https://mlsanity.com/paper/2402.07465/) (2024)
- [Efficient Neural Network Approaches for Conditional Optimal Transport with Applications in Bayesian Inference](https://mlsanity.com/paper/2310.16975/) (2023)
- [Learning the temporal evolution of multivariate densities via normalizing flows](https://mlsanity.com/paper/2107.13735/) (2021)
- [Solving time dependent Fokker-Planck equations via temporal normalizing flow](https://mlsanity.com/paper/2112.14012/) (2021)
- [Learning Green's Functions of Linear Reaction-Diffusion Equations with Application to Fast Numerical Solver](https://mlsanity.com/paper/2105.11045/) (2021)
- [Coupling parameter and particle dynamics for adaptive sampling in Neural Galerkin schemes](https://mlsanity.com/paper/2306.15630/) (2023)
- [Bounded KRnet and its applications to density estimation and approximation](https://mlsanity.com/paper/2305.09063/) (2023)
- [Neural Control of Parametric Solutions for High-dimensional Evolution PDEs](https://mlsanity.com/paper/2302.00045/) (2023)
- [Adaptive deep density approximation for fractional Fokker-Planck equations](https://mlsanity.com/paper/2210.14402/) (2022)
- [Adaptive deep density approximation for stochastic dynamical systems](https://mlsanity.com/paper/2405.02810/) (2024)
- [FlowKac: An Efficient Neural Fokker-Planck solver using Temporal Normalizing Flows and the Feynman-Kac Formula](https://mlsanity.com/paper/2503.11427/) (2025)
- [Simulating Fokker-Planck equations via mean field control of score-based normalizing flows](https://mlsanity.com/paper/2506.05723/) (2025)
- [Neural Galerkin Normalizing Flow for Transition Probability Density Functions of Diffusion Models](https://mlsanity.com/paper/2603.18907/) (2026)
- [A deep learning framework for jointly solving transient Fokker-Planck equations with arbitrary parameters and initial distributions](https://mlsanity.com/paper/2604.06001/) (2026)
- [Neural Galerkin Normalizing Flows for Bayesian Inference of Diffusions with Inaccessible Boundaries](https://mlsanity.com/paper/2606.04324/) (2026)

## Abstract (arXiv)

Solving Fokker-Planck equations (FPEs) for multiple initial conditions typically requires repeated computations, leading to substantial computational costs. In this work, we propose a transition-density-based operator learning method to efficiently approximate the solution operator of FPEs with various initial conditions. The core idea is to learn the transition probability density function (PDF) of the underlying stochastic differential equation (SDE), from which the solution associated with a new initial distribution can be obtained through the Chapman-Kolmogorov equation without retraining the model. A major challenge in learning the transition PDF lies in the singular behavior induced by the Dirac initial condition. To address it, we introduce a conditional normalizing flow whose base distribution is given by the explicit transition PDF of a linearized SDE. This base distribution captures the short-time behavior of the target transition PDF and allows the normalizing flow to learn a near-identity transformation at small times. We further incorporate a time-weighted loss function to stabilize training near the initial time and develop an importance-sampling strategy for evaluating solutions associated with general initial conditions. A variety of numerical experiments are presented to illustrate the effectiveness and robustness of the proposed method.