{ "id": "epj4xxw41k", "slug": "fokker-planck-model-machine-learning", "title": "Fokker–Planck model (machine learning)", "updated": "2026-09-30", "topic_path": [ { "id": "technology", "label": "Technology and the built world", "api_url": "https://www.edgepedia.org/api/v1/topics/technology" }, { "id": "technology.computing", "label": "Computing and digital systems", "api_url": "https://www.edgepedia.org/api/v1/topics/technology.computing" }, { "id": "technology.computing.ai", "label": "Artificial intelligence and data", "api_url": "https://www.edgepedia.org/api/v1/topics/technology.computing.ai" }, { "id": "technology.computing.ai.ml_computation", "label": "Machine learning and neural computation", "api_url": "https://www.edgepedia.org/api/v1/topics/technology.computing.ai.ml_computation" }, { "id": "technology.computing.ai.ml_methods", "label": "Machine learning methods", "api_url": "https://www.edgepedia.org/api/v1/topics/technology.computing.ai.ml_methods" } ], "geo": [ { "id": "geo.nongeo.t2001.technology.computing.ai.ml_methods", "label": "Non-geographic · 2001 to 2020: Machine learning methods", "api_url": "https://www.edgepedia.org/api/v1/geo/geo.nongeo.t2001.technology.computing.ai.ml_methods", "path": [ { "id": "geo.nongeo", "label": "Non-geographic", "api_url": "https://www.edgepedia.org/api/v1/geo/geo.nongeo" }, { "id": "geo.nongeo.t2001", "label": "Non-geographic · 2001 to 2020", "api_url": "https://www.edgepedia.org/api/v1/geo/geo.nongeo.t2001" }, { "id": "geo.nongeo.t2001.technology", "label": "Technology and the built world", "api_url": "https://www.edgepedia.org/api/v1/geo/geo.nongeo.t2001.technology" }, { "id": "geo.nongeo.t2001.technology.computing", "label": "Computing and digital systems", "api_url": "https://www.edgepedia.org/api/v1/geo/geo.nongeo.t2001.technology.computing" }, { "id": "geo.nongeo.t2001.technology.computing.ai", "label": "Artificial intelligence and data", "api_url": "https://www.edgepedia.org/api/v1/geo/geo.nongeo.t2001.technology.computing.ai" }, { "id": "geo.nongeo.t2001.technology.computing.ai.ml_computation", "label": "Machine learning and neural computation", "api_url": "https://www.edgepedia.org/api/v1/geo/geo.nongeo.t2001.technology.computing.ai.ml_computation" }, { "id": "geo.nongeo.t2001.technology.computing.ai.ml_methods", "label": "Machine learning methods", "api_url": "https://www.edgepedia.org/api/v1/geo/geo.nongeo.t2001.technology.computing.ai.ml_methods" } ] } ], "excerpt": "A Fokker–Planck model in machine learning uses the Fokker–Planck equation to describe how a probability density evolves over time, underpinning generative modeling, sampling, and density estimation.", "snippet": "A Fokker–Planck model in machine learning uses the Fokker–Planck equation to describe how a probability density evolves over time, underpinning generative modeling, sampling, and density estimation.", "node": "technology.computing.ai.ml_methods", "markdown": "# Fokker–Planck model (machine learning)\n\nA Fokker–Planck model in machine learning describes how the probability density of a stochastic process evolves over time using the [Fokker–Planck equation](https://www.edgepedia.org/fokker-planck-equation), and turns that description into tools for generative modeling, sampling, and density estimation. What a trained model actually outputs depends on the variant: some produce a sampler that maps reference noise to data samples without an explicit density[1](https://arxiv.org/pdf/2509.14575), some give direct access to the density, the probability current, and the entropy along the evolution[2](https://iopscience.iop.org/article/10.1088/2632-2153/ace2aa/pdf), and the diffusion-model family produces a score field (a gradient of log-density) from which both samplers and likelihoods follow.[3](https://doi.org/10.48550/arxiv.2011.13456)\n\n| Key fact | Value |\n|---|---|\n| Governing PDE (Itô form) | \\( \\partial_{t} \\rho_{t}(x) = -\\nabla \\cdot ( b_{t}(x) \\rho_{t}(x) ) + \\partial_{i}\\partial_{j} ( D_{t}(x) \\rho_{t}(x) ) \\)[2](https://iopscience.iop.org/article/10.1088/2632-2153/ace2aa/pdf) |\n| Score definition | \\( \\nabla_{x} \\log p(x) \\), approximated by a network \\( s_{\\theta} \\)[4](https://doi.org/10.48550/arxiv.1907.05600) |\n| Probability flow ODE | \\( dx = [ f(x,t) - \\tfrac{1}{2} g(t)^{2} \\nabla_{x} \\log p_{t}(x) ] \\, dt \\), same marginals as the SDE[3](https://doi.org/10.48550/arxiv.2011.13456) |\n| CIFAR-10 unconditional generation (SDE framework) | Inception score 9.89, FID 2.20, likelihood 2.99 bits/dim[3](https://doi.org/10.48550/arxiv.2011.13456) |\n| Sampling cost reduction | Over 90% fewer function evaluations with a black-box ODE solver, without visible quality loss[3](https://doi.org/10.48550/arxiv.2011.13456) |\n| Score FPE regularization (FP-Diffusion) | CIFAR-10 NLL 3.36 (VE) vs 3.61 vanilla, but FID 10.83 vs 3.33[5](https://doi.org/10.48550/arxiv.2210.04296) |\n| Direct PDE solving limit | Grid methods become infeasible at dimensionality \\( d \\) as small as five or six[2](https://iopscience.iop.org/article/10.1088/2632-2153/ace2aa/pdf) |\n\n## How it works\n\nThe Fokker–Planck equation is a partial differential equation for the temporal evolution of the probability density \\( \\rho_{t}(x) \\) of a state \\( x \\) in a stochastic or deterministic dynamical system.[2](https://iopscience.iop.org/article/10.1088/2632-2153/ace2aa/pdf) • [6](https://arxiv.org/html/2606.31576) For an Itô process with drift \\( b_{t} \\) and diffusion \\( D_{t} \\), the general equation is \\( \\partial_{t} \\rho_{t}(x) = -\\nabla \\cdot ( b_{t}(x) \\rho_{t}(x) ) + \\partial_{i}\\partial_{j} ( D_{t}(x) \\rho_{t}(x) ) \\); the flux form \\( \\partial_{t} \\rho_{t}(x) = -\\nabla \\cdot ( b_{t}(x) \\rho_{t}(x) - D_{t}(x) \\nabla \\rho_{t}(x) ) \\) displayed below holds when the diffusion matrix is constant, and describes the density changing by fluxes of probability into and out of each point.[2](https://iopscience.iop.org/article/10.1088/2632-2153/ace2aa/pdf)\n\nThe reverse of a noising process is another SDE whose drift depends on the score \\( \\nabla_{x} \\log p_{t}(x) \\)[7](https://d2l.smola.org/chapter_mdl-dynamics/mdl-fokker-planck-probability-flow.html), so learning the score at every time makes the density evolution reversible. Because there is a one-to-one mapping (up to a constant) between densities and their scores, the density Fokker–Planck equation has an equivalent PDE system for the scores, called the score Fokker–Planck equation.[5](https://doi.org/10.48550/arxiv.2210.04296) A deterministic alternative with the same marginals is the probability flow ODE, \\( dx = [ f(x,t) - \\tfrac{1}{2} g(t)^{2} \\nabla_{x} \\log p_{t}(x) ] \\, dt \\)[3](https://doi.org/10.48550/arxiv.2011.13456), equivalently a transport equation \\( \\partial_{t} \\rho_{t}^{*} = -\\nabla \\cdot ( v_{t}^{*} \\rho_{t}^{*} ) \\) with velocity \\( v_{t}^{*}(x) = b_{t}(x) - D_{t}(x) \\nabla \\log \\rho_{t}^{*}(x) \\).[2](https://iopscience.iop.org/article/10.1088/2632-2153/ace2aa/pdf) Along this flow the density is recoverable by a change of variables, \\( \\rho_{t}(x) = \\rho_{0}(X_{t,0}(x)) \\exp( -\\int_{0}^{t} \\nabla \\cdot v_{\\tau}(X_{t,\\tau}(x)) \\, d\\tau ) \\), which is what gives access to the density, probability current, and entropy that SDE trajectories alone do not provide.[2](https://iopscience.iop.org/article/10.1088/2632-2153/ace2aa/pdf)\n\n## How it is done\n\nThe standard pipeline has three stages. First, choose a forward SDE that gradually diffuses data into a simple prior; the 2015 diffusion procedure, for example, systematically and slowly destroys structure in the data distribution through an iterative forward diffusion process.[8](https://proceedings.mlr.press/v37/sohl-dickstein15) Second, train a score network \\( s_{\\theta}: \\mathbb{R}^{D} \\to \\mathbb{R}^{D} \\), conditioned on the time or noise level, to approximate \\( \\nabla_{x} \\log p_{t}(x) \\), the score of the perturbed marginal at each stage of the forward process, typically by denoising score matching applied to these perturbed distributions; the clean-data score \\( \\nabla_{x} \\log p_{\\text{data}}(x) \\) is only a limiting target.[4](https://doi.org/10.48550/arxiv.1907.05600) Third, sample by simulating the learned reverse dynamics. Plain [Langevin dynamics](https://www.edgepedia.org/langevin-dynamics) iterates \\( \\tilde{x}_{t} = \\tilde{x}_{t-1} + (\\varepsilon/2) \\nabla_{x} \\log p(\\tilde{x}_{t-1}) + \\sqrt{\\varepsilon} \\, z_{t} \\) with \\( z_{t} \\sim \\mathcal{N}(0, I) \\), and, under suitable regularity and ergodicity conditions, its distribution converges to \\( p(x) \\) as \\( \\varepsilon \\to 0 \\) while the total simulated time grows without bound; at fixed step size the unadjusted scheme carries a discretization bias.[4](https://doi.org/10.48550/arxiv.1907.05600)\n\nModern implementations replace plain Langevin with two special samplers: Predictor–Corrector samplers that combine SDE solvers with Langevin MCMC correction steps, and deterministic probability flow ODE samplers that additionally enable exact likelihood computation.[3](https://doi.org/10.48550/arxiv.2011.13456) [Discretization](https://www.edgepedia.org/discretization) matters for cost: with a black-box ODE solver (Dormand–Prince) and a larger error tolerance, the number of function evaluations can be reduced by over 90% without affecting the visual quality of samples.[3](https://doi.org/10.48550/arxiv.2011.13456)\n\n## Origin\n\nThe generative-modeling line begins with a 2015 ICML paper which described learning a reverse diffusion process that restores structure destroyed by a forward diffusion process, yielding a tractable generative model; the paper connects its procedure to Annealed Importance Sampling, described by Radford M. Neal in [Statistics and Computing](https://www.edgepedia.org/statistics-and-computing) in 2001, which uses a [Markov chain](https://www.edgepedia.org/markov-chain) slowly converting one distribution into another to compute ratios of normalizing constants.[8](https://proceedings.mlr.press/v37/sohl-dickstein15) • [9](https://doi.org/10.1023/a:1008923215028) Score-based generative modeling, combining score matching with Langevin dynamics, appears in Song and Ermon's 2019 paper[4](https://doi.org/10.48550/arxiv.1907.05600), and Denoising Diffusion Probabilistic Models by Ho, Jain, and Abbeel followed in 2020.[10](https://doi.org/10.48550/arxiv.2006.11239) According to Lai and colleagues, Song, Sohl-Dickstein, Kingma, Kumar, Ermon, and Poole unified denoising score matching and diffusion probabilistic models in 2020 via a continuous-time stochastic process driven by a forward SDE.[5](https://doi.org/10.48550/arxiv.2210.04296) • [3](https://doi.org/10.48550/arxiv.2011.13456) Older Fokker–Planck methodology of a similar kind exists in the time-series setting, rather than in generative modeling.[6](https://arxiv.org/html/2606.31576)\n\n## Variants\n\n**Score-based SDE models** train \\( s_{\\theta} \\) by (denoising) score matching and sample from the reverse-time SDE or its Predictor–Corrector and probability flow ODE samplers.[3](https://doi.org/10.48550/arxiv.2011.13456) Turning an SDE into an ODE and vice versa without changing the marginals enables deterministic sampling from a diffusion model and stochastic sampling from a deterministic flow model[6](https://arxiv.org/html/2606.31576); flow matching, set out by Lipman, Chen, Ben-Hamu, Nickel, and Le in 2022, trains such deterministic velocity fields directly[11](https://doi.org/10.48550/arxiv.2210.02747), and stochastic interpolants (Albergo, Boffi, and Vanden-Eijnden, 2023) unify flows and diffusions in one framework.[12](https://doi.org/10.48550/arxiv.2303.08797)\n\n**Direct Fokker–Planck solvers** attack the PDE itself. Neural parametric Fokker–Planck equations formulate the FPE as a system of ODEs on neural-network parameter space, derived as the constrained \\( L^{2} \\)-Wasserstein gradient flow of the KL divergence, and sample by pushing a reference distribution through a time-dependent map \\( T_{\\theta_{t}} \\).[13](https://ww3.math.ucla.edu/camreport/cam20-08.pdf) A mesh-free solver represents the solution of the Fokker–Planck equation directly with a neural network, addressing the unbounded, high-dimensional domain typical of the equation.[14](https://par.nsf.gov/biblio/10345329) FPNN solves 4–20 dimensional stationary Fokker–Planck equations for complex physical systems without labeled data or zero boundary conditions, incorporating boundary and normalization constraints as regularization.[15](https://proceedings.iclr.cc/paper_files/paper/2025/file/23aa2163dea287441ebebc1295d5b3fc-Paper-Conference.pdf) A weak-adversarial approach learns a neural pushforward map \\( F_{\\vartheta} \\) from a simple reference distribution via adversarial training on a weak formulation of the FPE in which the adjoint operator acts on test functions.[1](https://arxiv.org/pdf/2509.14575) Schrödinger bridge refinements learn a parametrized drift \\( u_{\\theta_{t}} \\) inside a diffusion sampler \\( dX_{t} = [ f_{t}(X_{t}) + \\sigma_{t} u_{\\theta_{t}}(X_{t}) ] \\, dt + \\sigma_{t} \\, dW_{t} \\) given a base drift \\( f_{t} \\), noise schedule \\( \\sigma_{t} \\), and source distribution.[16](https://papers.nips.cc/paper_files/paper/2025/file/174692c52dc84fad2b2e99dd8637ce6a-Paper-Conference.pdf)\n\n## Applications\n\nUnconditional image generation on CIFAR-10 and 1024×1024 images is the benchmark setting where the SDE-based Fokker–Planck machinery is best documented.[3](https://doi.org/10.48550/arxiv.2011.13456) In physics, the probability flow solution has been demonstrated on interacting particle systems, with the score modeled by a deep neural network learned on-the-fly without requiring samples from the target density[2](https://iopscience.iop.org/article/10.1088/2632-2153/ace2aa/pdf); this gives direct access to quantities that are challenging to estimate from stochastic trajectories, such as the probability current, the density itself, and its entropy.[2](https://iopscience.iop.org/article/10.1088/2632-2153/ace2aa/pdf) Stationary Fokker–Planck solvers such as FPNN target complex physical systems in 4–20 dimensions[15](https://proceedings.iclr.cc/paper_files/paper/2025/file/23aa2163dea287441ebebc1295d5b3fc-Paper-Conference.pdf), and consistency-model samplers are applied in simulation-based inference.[17](https://paulbuerkner.com/publications/pdf/2024__Schmitt_et_al__NeurIPS.pdf)\n\n## Limitations and alternatives\n\n**Score models can violate their own dynamics.** In practice, many existing pre-trained score models do not numerically satisfy the score Fokker–Planck equation, which motivates adding a score-FPE regularization term to the score matching objective.[5](https://doi.org/10.48550/arxiv.2210.04296)\n\n**ODE versus SDE samplers.** The published literature disagrees on quality: the SDE paper reports that with a black-box ODE solver the number of function evaluations can be reduced by over 90% without affecting visual sample quality[3](https://doi.org/10.48550/arxiv.2011.13456), while a later study states that \"Empirically, it has been reported that samplers based on ordinary differential equations (ODEs) are inferior to those based on stochastic differential equations (SDEs)\".[18](https://pmc.ncbi.nlm.nih.gov/articles/PMC12139524/) The disagreement has been at least partially resolved: a mathematical analysis of two limiting scenarios finds that the outcome depends on how score errors distribute over the generative time horizon, with ODE (zero-diffusion) samplers able to outperform when score errors occur only late in generation, and SDEs contracting mid-process errors and beating ODEs once discretization error is small.[18](https://pmc.ncbi.nlm.nih.gov/articles/PMC12139524/)\n\n**Dimensionality.** Standard grid-based numerical methods for the Fokker–Planck PDE become infeasible for \\( d \\) as small as five or six because computational complexity scales exponentially with \\( d \\); neural, mesh-free, and flow-based solvers exist precisely to avoid this.[2](https://iopscience.iop.org/article/10.1088/2632-2153/ace2aa/pdf) • [14](https://par.nsf.gov/biblio/10345329) Conversely, the [Monte Carlo](https://www.edgepedia.org/monte-carlo)/SDE approach only provides samples, so the density itself or the differential entropy requires interpolation methods that typically do not scale well to high dimension.[2](https://iopscience.iop.org/article/10.1088/2632-2153/ace2aa/pdf)\n\n**Training control.** Learning the score on external samples from the SDE does not control either direction of the KL divergence, whereas probability-flow-based self-consistent training controls the KL divergence from the learned solution to the target.[2](https://iopscience.iop.org/article/10.1088/2632-2153/ace2aa/pdf) Neural parametric Fokker–Planck methods are also explicitly distinct from Langevin Monte Carlo (LMC, MALA) methods, which target the stationary distribution of the SDE rather than the evolving density.[13](https://ww3.math.ucla.edu/camreport/cam20-08.pdf) Beyond that distinction, the published sources do not provide direct quantitative benchmarks against GANs, VAEs, normalizing flows, or MCMC, so no head-to-head comparison can be stated here.\n\n## References\n\n1. [Learning Neural Pushforward Samplers for Distributions from Fokker-Planck Equations by Weak Adversarial Training](https://arxiv.org/pdf/2509.14575)\n2. [Probability flow solution of the Fokker–Planck equation](https://iopscience.iop.org/article/10.1088/2632-2153/ace2aa/pdf)\n3. [Song, Yang and colleagues (2020). Score-Based Generative Modeling through Stochastic Differential Equations. arXiv (Cornell University).](https://doi.org/10.48550/arxiv.2011.13456)\n4. [Song, Yang, Ermon, Stefano (2019). Generative Modeling by Estimating Gradients of the Data Distribution. arXiv (Cornell University).](https://doi.org/10.48550/arxiv.1907.05600)\n5. [Lai, Chieh-Hsin and colleagues (2022). FP-Diffusion: Improving Score-based Diffusion Models by Enforcing the Underlying Score Fokker-Planck Equation. arXiv (Cornell University).](https://doi.org/10.48550/arxiv.2210.04296)\n6. [Introduction to Stochastic Differential Equations for Generative Machine Learning: A Variational Perspective](https://arxiv.org/html/2606.31576)\n7. [The Fokker–Planck Equation and Probability Flow – Dive into Deep Learning](https://d2l.smola.org/chapter_mdl-dynamics/mdl-fokker-planck-probability-flow.html)\n8. [Deep Unsupervised Learning using Nonequilibrium Thermodynamics](https://proceedings.mlr.press/v37/sohl-dickstein15)\n9. [Radford M. Neal (2001). Annealed importance sampling. Statistics and Computing.](https://doi.org/10.1023/a:1008923215028)\n10. [Ho, Jonathan, Jain, Ajay, Abbeel, Pieter (2020). Denoising Diffusion Probabilistic Models. arXiv (Cornell University).](https://doi.org/10.48550/arxiv.2006.11239)\n11. [Lipman, Yaron and colleagues (2022). Flow Matching for Generative Modeling. arXiv (Cornell University).](https://doi.org/10.48550/arxiv.2210.02747)\n12. [Albergo, Michael S., Boffi, Nicholas M., Vanden-Eijnden, Eric (2023). Stochastic Interpolants: A Unifying Framework for Flows and Diffusions. arXiv (Cornell University).](https://doi.org/10.48550/arxiv.2303.08797)\n13. [Neural Parametric Fokker-Planck Equations (UCLA CAM report)](https://ww3.math.ucla.edu/camreport/cam20-08.pdf)\n14. [A deep learning method for solving Fokker-Planck equations](https://par.nsf.gov/biblio/10345329)\n15. [Score-Based Free-Form Architectures for High-Dimensional Fokker-Planck Equations](https://proceedings.iclr.cc/paper_files/paper/2025/file/23aa2163dea287441ebebc1295d5b3fc-Paper-Conference.pdf)\n16. [Adjoint Schrödinger Bridge Sampler](https://papers.nips.cc/paper_files/paper/2025/file/174692c52dc84fad2b2e99dd8637ce6a-Paper-Conference.pdf)\n17. [Consistency Models for Scalable and Fast Simulation-Based Inference](https://paulbuerkner.com/publications/pdf/2024__Schmitt_et_al__NeurIPS.pdf)\n18. [Closing the ODE–SDE gap in score-based diffusion models through the Fokker–Planck equation](https://pmc.ncbi.nlm.nih.gov/articles/PMC12139524/)\n\n---\n*Topic: Encyclopedia › Technology and the built world › Computing and digital systems › Artificial intelligence and data › Machine learning and neural computation › Machine learning methods*\n\n*Initially written Sep 29, 2026 · Reviewed: Sep 30, 2026 · Edited: Sep 30, 2026 · Last review: Sep 30, 2026*\n\n*Copyright 2026 EdgeChat AI, a subsidiary of Biostate AI.*\n\nLicense: Edgepedia Community License 1.0, https://www.edgepedia.org/edgepedia/license\n", "same_as": [], "url": "https://www.edgepedia.org/fokker-planck-model-machine-learning", "markdown_url": "https://www.edgepedia.org/fokker-planck-model-machine-learning.md", "license": { "name": "Edgepedia Community License 1.0", "url": "https://www.edgepedia.org/edgepedia/license", "summary": "Free with credit, commercial use included. AI training is open to everyone. 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