Probability flow solution of the Fokker–Planck equation
The method of choice for integrating the time-dependent Fokker–Planck equation (FPE) in high-dimension is to generate samples from the solution via integration of the associated stochastic differential equation (SDE). Here, we study an alternative scheme based on integrating an ordinary differential...
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| Format: | Article |
| Language: | English |
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IOP Publishing
2023-01-01
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| Series: | Machine Learning: Science and Technology |
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| Online Access: | https://doi.org/10.1088/2632-2153/ace2aa |
| _version_ | 1797360946098733056 |
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| author | Nicholas M Boffi Eric Vanden-Eijnden |
| author_facet | Nicholas M Boffi Eric Vanden-Eijnden |
| author_sort | Nicholas M Boffi |
| collection | DOAJ |
| description | The method of choice for integrating the time-dependent Fokker–Planck equation (FPE) in high-dimension is to generate samples from the solution via integration of the associated stochastic differential equation (SDE). 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 Kullback–Leibler (KL) divergence from the learned solution to the target, while learning on external samples from the SDE does not control either direction of the KL divergence. Empirically, we consider several high-dimensional FPEs 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. |
| first_indexed | 2024-03-08T15:46:52Z |
| format | Article |
| id | doaj.art-83e1dbd791d245a4ac8dcd969831fe5b |
| institution | Directory Open Access Journal |
| issn | 2632-2153 |
| language | English |
| last_indexed | 2024-03-08T15:46:52Z |
| publishDate | 2023-01-01 |
| publisher | IOP Publishing |
| record_format | Article |
| series | Machine Learning: Science and Technology |
| spelling | doaj.art-83e1dbd791d245a4ac8dcd969831fe5b2024-01-09T10:34:36ZengIOP PublishingMachine Learning: Science and Technology2632-21532023-01-014303501210.1088/2632-2153/ace2aaProbability flow solution of the Fokker–Planck equationNicholas M Boffi0https://orcid.org/0000-0003-1336-7568Eric Vanden-Eijnden1Courant Institute of Mathematical Sciences, New York University , New York, NY 10012, United States of AmericaCourant Institute of Mathematical Sciences, New York University , New York, NY 10012, United States of AmericaThe method of choice for integrating the time-dependent Fokker–Planck equation (FPE) in high-dimension is to generate samples from the solution via integration of the associated stochastic differential equation (SDE). 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 Kullback–Leibler (KL) divergence from the learned solution to the target, while learning on external samples from the SDE does not control either direction of the KL divergence. Empirically, we consider several high-dimensional FPEs 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.https://doi.org/10.1088/2632-2153/ace2aaFokker–Planck equationstatistical physicsstochastic dynamicshigh-dimensional scientific computingdiffusion models |
| spellingShingle | Nicholas M Boffi Eric Vanden-Eijnden Probability flow solution of the Fokker–Planck equation Machine Learning: Science and Technology Fokker–Planck equation statistical physics stochastic dynamics high-dimensional scientific computing diffusion models |
| title | Probability flow solution of the Fokker–Planck equation |
| title_full | Probability flow solution of the Fokker–Planck equation |
| title_fullStr | Probability flow solution of the Fokker–Planck equation |
| title_full_unstemmed | Probability flow solution of the Fokker–Planck equation |
| title_short | Probability flow solution of the Fokker–Planck equation |
| title_sort | probability flow solution of the fokker planck equation |
| topic | Fokker–Planck equation statistical physics stochastic dynamics high-dimensional scientific computing diffusion models |
| url | https://doi.org/10.1088/2632-2153/ace2aa |
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