{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,7,15]],"date-time":"2026-07-15T18:50:31Z","timestamp":1784141431578,"version":"3.55.0"},"reference-count":67,"publisher":"MDPI AG","issue":"8","license":[{"start":{"date-parts":[[2020,7,22]],"date-time":"2020-07-22T00:00:00Z","timestamp":1595376000000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"DOI":"10.13039\/501100001659","name":"Deutsche Forschungsgemeinschaft","doi-asserted-by":"publisher","award":["SFB1294\/1 - 442 318763901"],"award-info":[{"award-number":["SFB1294\/1 - 442 318763901"]}],"id":[{"id":"10.13039\/501100001659","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Entropy"],"abstract":"<jats:p>Fokker\u2013Planck 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\u2013Planck 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\u2013Planck equations in low and moderate dimensions. The proposed gradient\u2013log\u2013density estimator is also of independent interest, for example, in the context of optimal control.<\/jats:p>","DOI":"10.3390\/e22080802","type":"journal-article","created":{"date-parts":[[2020,7,22]],"date-time":"2020-07-22T07:31:28Z","timestamp":1595403088000},"page":"802","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":50,"title":["Interacting Particle Solutions of Fokker\u2013Planck Equations Through Gradient\u2013Log\u2013Density Estimation"],"prefix":"10.3390","volume":"22","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-3553-8658","authenticated-orcid":false,"given":"Dimitra","family":"Maoutsa","sequence":"first","affiliation":[{"name":"Artificial Intelligence Group, Technische Universit\u00e4t Berlin, Marchstra\u00dfe 23, 10587 Berlin, Germany"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-5336-8904","authenticated-orcid":false,"given":"Sebastian","family":"Reich","sequence":"additional","affiliation":[{"name":"Institute of Mathematics, University of Potsdam, Karl-Liebknecht-Str. 24\/25, 14476 Potsdam, Germany"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-2856-7589","authenticated-orcid":false,"given":"Manfred","family":"Opper","sequence":"additional","affiliation":[{"name":"Artificial Intelligence Group, Technische Universit\u00e4t Berlin, Marchstra\u00dfe 23, 10587 Berlin, Germany"}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"1968","published-online":{"date-parts":[[2020,7,22]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"12795","DOI":"10.1073\/pnas.162041399","article-title":"Intrinsic and extrinsic contributions to stochasticity in gene expression","volume":"99","author":"Swain","year":"2002","journal-title":"Proc. Natl. Acad. Sci. USA"},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"061138","DOI":"10.1103\/PhysRevE.77.061138","article-title":"Intrinsic and extrinsic noise effects on phase transitions of network models with applications to swarming systems","volume":"77","author":"Pimentel","year":"2008","journal-title":"Phys. Rev. E"},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"12167","DOI":"10.1073\/pnas.1018832108","article-title":"Separating intrinsic from extrinsic fluctuations in dynamic biological systems","volume":"108","author":"Hilfinger","year":"2011","journal-title":"Proc. Natl. Acad. Sci. USA"},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"1183","DOI":"10.1126\/science.1070919","article-title":"Stochastic gene expression in a single cell","volume":"297","author":"Elowitz","year":"2002","journal-title":"Science"},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"131","DOI":"10.1016\/S0166-2236(99)01521-0","article-title":"Channel noise in neurons","volume":"23","author":"White","year":"2000","journal-title":"Trends Neurosci."},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"861","DOI":"10.1038\/nature04281","article-title":"Origins of extrinsic variability in eukaryotic gene expression","volume":"439","author":"Volfson","year":"2006","journal-title":"Nature"},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"811","DOI":"10.1016\/j.neuroscience.2003.08.027","article-title":"Synaptic background noise controls the input\/output characteristics of single cells in an in vitro model of in vivo activity","volume":"122","author":"Fellous","year":"2003","journal-title":"Neuroscience"},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"093001","DOI":"10.1088\/1751-8121\/aa54d9","article-title":"Approximation and inference methods for stochastic biochemical kinetics\u2014A tutorial review","volume":"50","author":"Schnoerr","year":"2017","journal-title":"J. Phys. Math. Theor."},{"key":"ref_9","first-page":"245","article-title":"The expansion of the master equation","volume":"34","year":"1976","journal-title":"Adv. Chem. Phys."},{"key":"ref_10","first-page":"195","article-title":"Passage from an initial unstable state to a final stable state","volume":"46","author":"Suzuki","year":"1981","journal-title":"Adv. Chem. Phys."},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"223","DOI":"10.1103\/RevModPhys.70.223","article-title":"Stochastic resonance","volume":"70","author":"Gammaitoni","year":"1998","journal-title":"Rev. Mod. Phys."},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"10","DOI":"10.3402\/tellusa.v34i1.10782","article-title":"Stochastic resonance in climatic change","volume":"34","author":"Benzi","year":"1982","journal-title":"Tellus"},{"key":"ref_13","doi-asserted-by":"crossref","unstructured":"Horsthemke, W. (1984). Noise induced transitions. Non-Equilibrium Dynamics in Chemical Systems, Springer.","DOI":"10.1007\/978-3-642-70196-2_23"},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"171","DOI":"10.1016\/S0034-4877(12)60037-8","article-title":"External noise effects in doped semiconductors operating under sub-THz signals","volume":"70","author":"Adorno","year":"2012","journal-title":"Rep. Math. Phys."},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"058102","DOI":"10.1103\/PhysRevLett.111.058102","article-title":"Extrinsic noise driven phenotype switching in a self-regulating gene","volume":"111","author":"Assaf","year":"2013","journal-title":"Phys. Rev. Lett."},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"910","DOI":"10.1016\/j.cell.2011.01.030","article-title":"Cellular decision making and biological noise: From microbes to mammals","volume":"144","author":"Collins","year":"2011","journal-title":"Cell"},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"805","DOI":"10.1111\/j.1469-7793.1999.0805s.x","article-title":"All thalamocortical neurones possess a T-type Ca2+ \u2018window\u2019current that enables the expression of bistability-mediated activities","volume":"517","author":"Hughes","year":"1999","journal-title":"J. Physiol."},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"769","DOI":"10.1152\/jn.1967.30.4.769","article-title":"Phase-locked response to low-frequency tones in single auditory nerve fibers of the squirrel monkey","volume":"30","author":"Rose","year":"1967","journal-title":"J. Neurophysiol."},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"473","DOI":"10.1007\/BF00154530","article-title":"Solar variability and stochastic effects on climate","volume":"74","author":"Nicolis","year":"1981","journal-title":"Sol. Phys."},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"1","DOI":"10.3402\/tellusa.v34i1.10781","article-title":"Stochastic aspects of climatic transitions\u2013response to a periodic forcing","volume":"34","author":"Nicolis","year":"1982","journal-title":"Tellus"},{"key":"ref_21","doi-asserted-by":"crossref","unstructured":"Risken, H. (1996). Fokker-Planck equation. The Fokker-Planck Equation, Springer.","DOI":"10.1007\/978-3-642-61544-3"},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"323","DOI":"10.1103\/RevModPhys.17.323","article-title":"On the theory of the Brownian motion II","volume":"17","author":"Wang","year":"1945","journal-title":"Rev. Mod. Phys."},{"key":"ref_23","doi-asserted-by":"crossref","unstructured":"S\u00e4rkk\u00e4, S., and Solin, A. (2019). Applied Stochastic Differential Equations, Cambridge University Press.","DOI":"10.1017\/9781108186735"},{"key":"ref_24","doi-asserted-by":"crossref","first-page":"297","DOI":"10.1063\/1.481811","article-title":"The chemical Langevin equation","volume":"113","author":"Gillespie","year":"2000","journal-title":"J. Chem. Phys."},{"key":"ref_25","doi-asserted-by":"crossref","first-page":"164109","DOI":"10.1063\/1.3380661","article-title":"Fast stochastic simulation of biochemical reaction systems by alternative formulations of the chemical Langevin equation","volume":"132","author":"Melykuti","year":"2010","journal-title":"J. Chem. Phys."},{"key":"ref_26","unstructured":"Schadschneider, A., Chowdhury, D., and Nishinari, K. (2010). Stochastic Transport in Complex Systems: From Molecules to Vehicles, Elsevier."},{"key":"ref_27","doi-asserted-by":"crossref","first-page":"445","DOI":"10.1007\/BF02716786","article-title":"Solution of Fokker-Planck equation by finite element and finite difference methods for nonlinear systems","volume":"31","author":"Kumar","year":"2006","journal-title":"Sadhana"},{"key":"ref_28","doi-asserted-by":"crossref","unstructured":"Brics, M., Kaupuzs, J., and Mahnke, R. (2013). How to solve Fokker-Planck equation treating mixed eigenvalue spectrum?. arXiv.","DOI":"10.5488\/CMP.16.13002"},{"key":"ref_29","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1016\/0021-9991(70)90001-X","article-title":"A practical difference scheme for Fokker-Planck equations","volume":"6","author":"Chang","year":"1970","journal-title":"J. Comput. Phys."},{"key":"ref_30","doi-asserted-by":"crossref","unstructured":"Pichler, L., Masud, A., and Bergman, L.A. (2013). Numerical solution of the Fokker\u2013Planck equation by finite difference and finite element methods\u2014A comparative study. Computational Methods in Stochastic Dynamics, Springer.","DOI":"10.1007\/978-94-007-5134-7_5"},{"key":"ref_31","doi-asserted-by":"crossref","first-page":"219","DOI":"10.1002\/num.1690040305","article-title":"Numerical solution of the Fokker Planck equation using moving finite elements","volume":"4","author":"Harrison","year":"1988","journal-title":"Numer. Methods Partial Differ. Equ."},{"key":"ref_32","doi-asserted-by":"crossref","first-page":"291","DOI":"10.1006\/jcph.1994.1101","article-title":"Implicit and conservative difference scheme for the Fokker-Planck equation","volume":"112","author":"Epperlein","year":"1994","journal-title":"J. Comput. Phys."},{"key":"ref_33","doi-asserted-by":"crossref","unstructured":"Leimkuhler, B., and Reich, S. (2004). Simulating Hamiltonian Dynamics, Cambridge University Press.","DOI":"10.1017\/CBO9780511614118"},{"key":"ref_34","doi-asserted-by":"crossref","first-page":"242","DOI":"10.1016\/j.jcp.2017.10.022","article-title":"Efficient statistically accurate algorithms for the Fokker\u2013Planck equation in large dimensions","volume":"354","author":"Chen","year":"2018","journal-title":"J. Comput. Phys."},{"key":"ref_35","unstructured":"Lin, Y., and Cai, G. (1995). Probabilistic Structural Dynamics: Advanced Theory and Applications, McGraw-Hill."},{"key":"ref_36","unstructured":"Roberts, J.B., and Spanos, P.D. (2003). Random Vibration and Statistical Linearization, Courier Corporation."},{"key":"ref_37","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1016\/S0266-8920(02)00037-1","article-title":"Equivalent linearization and Monte Carlo simulation in stochastic dynamics","volume":"18","author":"Proppe","year":"2003","journal-title":"Probab. Eng. Mech."},{"key":"ref_38","unstructured":"Grigoriu, M. (2013). Stochastic Calculus: Applications in Science and Engineering, Springer Science & Business Media."},{"key":"ref_39","doi-asserted-by":"crossref","unstructured":"\u00d8Ksendal, B. (2003). Stochastic Differential Equations, Springer.","DOI":"10.1007\/978-3-642-14394-6"},{"key":"ref_40","unstructured":"Kroese, D.P., Taimre, T., and Botev, Z.I. (2013). Handbook of Monte Carlo Methods, John Wiley &; Sons."},{"key":"ref_41","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1007\/s00526-019-1486-3","article-title":"A blob method for diffusion","volume":"58","author":"Carrillo","year":"2019","journal-title":"Calc. Var. Partial Differ. Equ."},{"key":"ref_42","first-page":"236","article-title":"Discrete gradients for computational Bayesian inference","volume":"6","author":"Pathiraja","year":"2019","journal-title":"J. Comp. Dyn."},{"key":"ref_43","unstructured":"Reich, S., and Weissmann, S. (2019). Fokker-Planck particle systems for Bayesian inference: Computational approaches. arXiv."},{"key":"ref_44","unstructured":"Liu, Q., Lee, J., and Jordan, M. (2016, January 19\u201324). A kernelized Stein discrepancy for goodness-of-fit tests. Proceedings of the International Conference on Machine Learning, New York, NY, USA."},{"key":"ref_45","unstructured":"Taghvaei, A., and Mehta, P.G. (2019). Accelerated flow for probability distributions. arXiv."},{"key":"ref_46","doi-asserted-by":"crossref","first-page":"82","DOI":"10.3390\/e13010082","article-title":"Entropy production: Its role in non-equilibrium thermodynamics","volume":"13","author":"Velasco","year":"2011","journal-title":"Entropy"},{"key":"ref_47","first-page":"695","article-title":"Estimation of non-normalized statistical models by score matching","volume":"6","year":"2005","journal-title":"J. Mach. Learn. Res."},{"key":"ref_48","unstructured":"Li, Y., and Turner, R.E. (2017). Gradient estimators for implicit models. arXiv."},{"key":"ref_49","unstructured":"Shi, J., Sun, S., and Zhu, J. (2018). A spectral approach to gradient estimation for implicit distributions. arXiv."},{"key":"ref_50","doi-asserted-by":"crossref","unstructured":"Tom\u00e9, T., and De Oliveira, M.J. (2015). Stochastic Dynamics and Irreversibility, Springer.","DOI":"10.1007\/978-3-319-11770-6"},{"key":"ref_51","doi-asserted-by":"crossref","unstructured":"Shawe-Taylor, J., and Cristianini, N. (2004). Kernel Methods for Pattern Analysis, Cambridge University Press.","DOI":"10.1017\/CBO9780511809682"},{"key":"ref_52","doi-asserted-by":"crossref","unstructured":"Scholkopf, B., and Smola, A.J. (2001). Learning with Kernels: Support Vector Machines, Regularization, Optimization, and Beyond, MIT Press.","DOI":"10.7551\/mitpress\/4175.001.0001"},{"key":"ref_53","unstructured":"Sutherland, D.J., Strathmann, H., Arbel, M., and Gretton, A. (2017). Efficient and principled score estimation with Nystr\u00f6m kernel exponential families. arXiv."},{"key":"ref_54","doi-asserted-by":"crossref","unstructured":"Rasmussen, C.E. (2003). Gaussian Processes in Machine Learning, Springer. Summer School on Machine Learning.","DOI":"10.1007\/978-3-540-28650-9_4"},{"key":"ref_55","unstructured":"Liu, Q., and Wang, D. (2016, January 5\u201310). Stein variational gradient descent: A general purpose Bayesian inference algorithm. Proceedings of the Advances in Neural Information Processing Systems, Barcelona, Spain."},{"key":"ref_56","unstructured":"Liu, Q. (2017, January 4\u20139). Stein variational gradient descent as gradient flow. Proceedings of the Advances in Neural Information Processing Systems, Long Beach, CA, USA."},{"key":"ref_57","doi-asserted-by":"crossref","unstructured":"Garbuno-Inigo, A., N\u00fcsken, N., and Reich, S. (2019). Affine invariant interacting Langevin dynamics for Bayesian inference. arXiv.","DOI":"10.1137\/19M1304891"},{"key":"ref_58","doi-asserted-by":"crossref","first-page":"101","DOI":"10.1081\/PDE-100002243","article-title":"The geometry of dissipative evolution equations: The porous medium equation","volume":"26","author":"Otto","year":"2001","journal-title":"Commun. Partial Differ. Equ."},{"key":"ref_59","unstructured":"Villani, C. (2008). Optimal Transport: Old and New, Springer Science & Business Media."},{"key":"ref_60","unstructured":"Frogner, C., and Poggio, T. (2018). Approximate inference with Wasserstein gradient flows. arXiv."},{"key":"ref_61","doi-asserted-by":"crossref","unstructured":"Caluya, K., and Halder, A. (2019). Gradient flow algorithms for density propagation in stochastic systems. IEEE Trans. Autom. Control.","DOI":"10.1109\/TAC.2019.2951348"},{"key":"ref_62","doi-asserted-by":"crossref","first-page":"083404","DOI":"10.1088\/1742-5468\/2016\/08\/083404","article-title":"Variational estimation of the drift for stochastic differential equations from the empirical density","volume":"2016","author":"Batz","year":"2016","journal-title":"J. Stat. Mech. Theory Exp."},{"key":"ref_63","doi-asserted-by":"crossref","first-page":"81","DOI":"10.1016\/j.physd.2007.03.011","article-title":"Computing ergodic limits for Langevin equations","volume":"229","author":"Milstein","year":"2007","journal-title":"Phys. D Nonlinear Phenom."},{"key":"ref_64","unstructured":"Saremi, S., Mehrjou, A., Sch\u00f6lkopf, B., and Hyv\u00e4rinen, A. (2018). Deep energy estimator networks. arXiv."},{"key":"ref_65","doi-asserted-by":"crossref","unstructured":"Reich, S., and Cotter, C. (2015). Probabilistic Forecasting and Bayesian Data Assimilation, Cambridge University Press.","DOI":"10.1017\/CBO9781107706804"},{"key":"ref_66","doi-asserted-by":"crossref","first-page":"130","DOI":"10.1175\/1520-0469(1963)020<0130:DNF>2.0.CO;2","article-title":"Deterministic nonperiodic flow","volume":"20","author":"Lorenz","year":"1963","journal-title":"J. Atmos. Sci."},{"key":"ref_67","unstructured":"Bobkov, S., and Ledoux, M. (2014). One-Dimensional Empirical Measures, Order Statistics and Kantorovich Transport Distances, Amer Mathematical Society."}],"container-title":["Entropy"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/1099-4300\/22\/8\/802\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T09:50:40Z","timestamp":1760176240000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/1099-4300\/22\/8\/802"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2020,7,22]]},"references-count":67,"journal-issue":{"issue":"8","published-online":{"date-parts":[[2020,8]]}},"alternative-id":["e22080802"],"URL":"https:\/\/doi.org\/10.3390\/e22080802","relation":{},"ISSN":["1099-4300"],"issn-type":[{"value":"1099-4300","type":"electronic"}],"subject":[],"published":{"date-parts":[[2020,7,22]]}}}