Interacting particle solutions of Fokker–Planck equations through gradient–log–density estimation
- Fokker-Planck equations are extensively employed in various scientific fields as they characterise the behaviour of stochastic systems at the level of probability density functions. Although broadly used, they allow for analytical treatment only in limited settings, and often it is inevitable to resort to numerical solutions. Here, we develop a computational approach for simulating the time evolution of Fokker-Planck solutions in terms of a mean field limit of an interacting particle system. The interactions between particles are determined by the gradient of the logarithm of the particle density, approximated here by a novel statistical estimator. The performance of our method shows promising results, with more accurate and less fluctuating statistics compared to direct stochastic simulations of comparable particle number. Taken together, our framework allows for effortless and reliable particle-based simulations of Fokker-Planck equations in low and moderate dimensions. The proposed gradient-log-density estimator is also ofFokker-Planck equations are extensively employed in various scientific fields as they characterise the behaviour of stochastic systems at the level of probability density functions. Although broadly used, they allow for analytical treatment only in limited settings, and often it is inevitable to resort to numerical solutions. Here, we develop a computational approach for simulating the time evolution of Fokker-Planck solutions in terms of a mean field limit of an interacting particle system. The interactions between particles are determined by the gradient of the logarithm of the particle density, approximated here by a novel statistical estimator. The performance of our method shows promising results, with more accurate and less fluctuating statistics compared to direct stochastic simulations of comparable particle number. Taken together, our framework allows for effortless and reliable particle-based simulations of Fokker-Planck equations in low and moderate dimensions. The proposed gradient-log-density estimator is also of independent interest, for example, in the context of optimal control.…
Author details: | Dimitra MaoutsaORCiD, Sebastian ReichORCiDGND, Manfred OpperORCiDGND |
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DOI: | https://doi.org/10.3390/e22080802 |
ISSN: | 1099-4300 |
Pubmed ID: | https://pubmed.ncbi.nlm.nih.gov/33286573 |
Title of parent work (English): | Entropy |
Publisher: | MDPI |
Place of publishing: | Basel |
Publication type: | Article |
Language: | English |
Date of first publication: | 2020/07/22 |
Publication year: | 2020 |
Release date: | 2023/03/30 |
Tag: | Fokker-Planck equation; gradient flow; interacting particles; multiplicative noise; stochastic differential equations; stochastic systems |
Volume: | 22 |
Issue: | 8 |
Article number: | 802 |
Number of pages: | 35 |
Funding institution: | Deutsche Forschungsgemeinschaft (DFG)German Research Foundation (DFG); [SFB1294/1-318763901] |
Organizational units: | Mathematisch-Naturwissenschaftliche Fakultät / Institut für Mathematik |
DDC classification: | 5 Naturwissenschaften und Mathematik / 51 Mathematik / 510 Mathematik |
Peer review: | Referiert |
Publishing method: | Open Access / Gold Open-Access |
DOAJ gelistet | |
License (German): | CC-BY - Namensnennung 4.0 International |