# Probability flow solution of the Fokker-Planck equation

arXiv: https://arxiv.org/abs/2206.04642 · HTML: https://mlsanity.com/paper/2206.04642/

## Key facts

- Cited by 47 papers in the map (data through 14 Oct 2026).
- Builds on 29 of its 64 references that are in the map.
- First posted on arXiv 9 Jun 2022.

## Builds on (most cited first, 29 of 29)

- [Adam: A Method for Stochastic Optimization](https://mlsanity.com/paper/1412.6980/) (2014)
- [Score-Based Generative Modeling through Stochastic Differential Equations](https://mlsanity.com/paper/2011.13456/) (2020)
- [Batch Normalization: Accelerating Deep Network Training by Reducing Internal Covariate Shift](https://mlsanity.com/paper/1502.03167/) (2015)
- [Generative Modeling by Estimating Gradients of the Data Distribution](https://mlsanity.com/paper/1907.05600/) (2019)
- [Variational Inference with Normalizing Flows](https://mlsanity.com/paper/1505.05770/) (2015)
- [Variational Inference: A Review for Statisticians](https://mlsanity.com/paper/1601.00670/) (2016)
- [Normalizing Flows for Probabilistic Modeling and Inference](https://mlsanity.com/paper/1912.02762/) (2019)
- [Improved Techniques for Training Score-Based Generative Models](https://mlsanity.com/paper/2006.09011/) (2020)
- [DGM: A deep learning algorithm for solving partial differential equations](https://mlsanity.com/paper/1708.07469/) (2017)
- [Diffusion Schr"odinger Bridge with Applications to Score-Based Generative Modeling](https://mlsanity.com/paper/2106.01357/) (2021)
- [The Deep Ritz method: A deep learning-based numerical algorithm for solving variational problems](https://mlsanity.com/paper/1710.00211/) (2017)
- [Solving high-dimensional partial differential equations using deep learning](https://mlsanity.com/paper/1707.02568/) (2017)
- [Stein Variational Gradient Descent: A General Purpose Bayesian Inference Algorithm](https://mlsanity.com/paper/1608.04471/) (2016)
- [How to Train Your Energy-Based Models](https://mlsanity.com/paper/2101.03288/) (2021)
- [Advances in Variational Inference](https://mlsanity.com/paper/1711.05597/) (2017)
- [Score-Based Generative Modeling with Critically-Damped Langevin Diffusion](https://mlsanity.com/paper/2112.07068/) (2021)
- [Symbolic Music Generation with Diffusion Models](https://mlsanity.com/paper/2103.16091/) (2021)
- [Stein Variational Gradient Descent as Gradient Flow](https://mlsanity.com/paper/1704.07520/) (2017)
- [Convex Potential Flows: Universal Probability Distributions with Optimal Transport and Convex Optimization](https://mlsanity.com/paper/2012.05942/) (2020)
- [Interacting particle solutions of Fokker-Planck equations through gradient-log-density estimation](https://mlsanity.com/paper/2006.00702/) (2020)
- [Neural Galerkin Schemes with Active Learning for High-Dimensional Evolution Equations](https://mlsanity.com/paper/2203.01360/) (2022)
- [Scaling limit of the Stein variational gradient descent: the mean field regime](https://mlsanity.com/paper/1805.04035/) (2018)
- [An invitation to sequential Monte Carlo samplers](https://mlsanity.com/paper/2007.11936/) (2020)
- [Provable Bayesian Inference via Particle Mirror Descent](https://mlsanity.com/paper/1506.03101/) (2015)
- [Self-Consistency of the Fokker-Planck Equation](https://mlsanity.com/paper/2206.00860/) (2022)
- [An introduction to sampling via measure transport](https://mlsanity.com/paper/1602.05023/) (2016)
- [Stein Variational Gradient Descent as Moment Matching](https://mlsanity.com/paper/1810.11693/) (2018)
- [Variational Particle Approximations](https://mlsanity.com/paper/1402.5715/) (2014)
- [A stochastic version of Stein Variational Gradient Descent for efficient sampling](https://mlsanity.com/paper/1902.03394/) (2019)

## Built on by (most cited first, 30 of 47)

- [Building Normalizing Flows with Stochastic Interpolants](https://mlsanity.com/paper/2209.15571/) (2022)
- [Stochastic Interpolants: A Unifying Framework for Flows and Diffusions](https://mlsanity.com/paper/2303.08797/) (2023)
- [Improved Analysis of Score-based Generative Modeling: User-Friendly Bounds under Minimal Smoothness Assumptions](https://mlsanity.com/paper/2211.01916/) (2022)
- [The probability flow ODE is provably fast](https://mlsanity.com/paper/2305.11798/) (2023)
- [Flow map matching with stochastic interpolants: A mathematical framework for consistency models](https://mlsanity.com/paper/2406.07507/) (2024)
- [How to build a consistency model: Learning flow maps via self-distillation](https://mlsanity.com/paper/2505.18825/) (2025)
- [Convergence Analysis of Probability Flow ODE for Score-based Generative Models](https://mlsanity.com/paper/2404.09730/) (2024)
- [Accelerating Diffusion Models with Parallel Sampling: Inference at Sub-Linear Time Complexity](https://mlsanity.com/paper/2405.15986/) (2024)
- [Normalizing flow neural networks by JKO scheme](https://mlsanity.com/paper/2212.14424/) (2022)
- [Sampling via Gradient Flows in the Space of Probability Measures](https://mlsanity.com/paper/2310.03597/) (2023)
- [Self-Consistent Velocity Matching of Probability Flows](https://mlsanity.com/paper/2301.13737/) (2023)
- [Gradient Flows for Sampling: Mean-Field Models, Gaussian Approximations and Affine Invariance](https://mlsanity.com/paper/2302.11024/) (2023)
- [Deep learning probability flows and entropy production rates in active matter](https://mlsanity.com/paper/2309.12991/) (2023)
- [Score-based Transport Modeling for Mean-Field Fokker-Planck Equations](https://mlsanity.com/paper/2305.03729/) (2023)
- [On the Equivalence of Consistency-Type Models: Consistency Models, Consistent Diffusion Models, and Fokker-Planck Regularization](https://mlsanity.com/paper/2306.00367/) (2023)
- [Parametric model reduction of mean-field and stochastic systems via higher-order action matching](https://mlsanity.com/paper/2410.12000/) (2024)
- [Lipschitz-regularized gradient flows and generative particle algorithms for high-dimensional scarce data](https://mlsanity.com/paper/2210.17230/) (2022)
- [Score-based Neural Ordinary Differential Equations for Computing Mean Field Control Problems](https://mlsanity.com/paper/2409.16471/) (2024)
- [A score-based particle method for homogeneous Landau equation](https://mlsanity.com/paper/2405.05187/) (2024)
- [Almost Linear Convergence under Minimal Score Assumptions: Quantized Transition Diffusion](https://mlsanity.com/paper/2505.21892/) (2025)
- [Generative Modeling from Black-box Corruptions via Self-Consistent Stochastic Interpolants](https://mlsanity.com/paper/2512.10857/) (2025)
- [Entropy-dissipation Informed Neural Network for McKean-Vlasov Type PDEs](https://mlsanity.com/paper/2303.11205/) (2023)
- [Transport based particle methods for the Fokker-Planck-Landau equation](https://mlsanity.com/paper/2405.10392/) (2024)
- [Score-based deterministic density sampling](https://mlsanity.com/paper/2504.18130/) (2025)
- [Deep Stochastic Mechanics](https://mlsanity.com/paper/2305.19685/) (2023)
- [FlowKac: An Efficient Neural Fokker-Planck solver using Temporal Normalizing Flows and the Feynman-Kac Formula](https://mlsanity.com/paper/2503.11427/) (2025)
- [Integration Matters for Learning PDEs with Backward SDEs](https://mlsanity.com/paper/2505.01078/) (2025)
- [Do Heavy Tails Help Diffusion? On the Subtle Trade-off Between Initialization and Training](https://mlsanity.com/paper/2605.13175/) (2026)
- [Score-fPINN: Fractional Score-Based Physics-Informed Neural Networks for High-Dimensional Fokker-Planck-Levy Equations](https://mlsanity.com/paper/2406.11676/) (2024)
- [The Effect of Stochasticity in Score-Based Diffusion Sampling: a KL Divergence Analysis](https://mlsanity.com/paper/2506.11378/) (2025)

## Abstract (arXiv)

The method of choice for integrating the time-dependent Fokker-Planck equation in high-dimension is to generate samples from the solution via integration of the associated stochastic differential equation. Here, we study an alternative scheme based on integrating an ordinary differential equation that describes the flow of probability. Acting as a transport map, this equation deterministically pushes samples from the initial density onto samples from the solution at any later time. Unlike integration of the stochastic dynamics, the method has the advantage of giving direct access to quantities that are challenging to estimate from trajectories alone, such as the probability current, the density itself, and its entropy. The probability flow equation depends on the gradient of the logarithm of the solution (its "score"), and so is a-priori unknown. To resolve this dependence, we model the score with a deep neural network that is learned on-the-fly by propagating a set of samples according to the instantaneous probability current. We show theoretically that the proposed approach controls the KL divergence from the learned solution to the target, while learning on external samples from the stochastic differential equation does not control either direction of the KL divergence. Empirically, we consider several high-dimensional Fokker-Planck equations from the physics of interacting particle systems. We find that the method accurately matches analytical solutions when they are available as well as moments computed via Monte-Carlo when they are not. Moreover, the method offers compelling predictions for the global entropy production rate that out-perform those obtained from learning on stochastic trajectories, and can effectively capture non-equilibrium steady-state probability currents over long time intervals.