Interacting Particle Solutions of Fokker–Planck Equations Through Gradient–Log–Density Estimation

By Dimitra Maoutsa, Sebastian Reich, Manfred Opper. 2020

139 citations recorded by the source corpus.

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 of independent interest, for example, in the context of optimal control.

Subfields: Statistical and Nonlinear Physics, Statistics and Probability, Probabilistic & Bayesian Methods

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Citations in the corpus

Score-Based Generative Modeling through Stochastic Differential Equations (2020, positive): “The idea of probability ﬂow ODE is inspired by Maoutsa et al. (2020), and one can ﬁnd the derivation of a simpliﬁed case therein.”
Flow Matching for Generative Modeling (2022, positive): “While existing works make use of a connection between diffusion processes and continuous normalizing flows with the same probability path (Maoutsa et al., 2020b; Song et al., 2020b; 2021), our work allows us to generalize beyond the class of probability paths modeled by simple diffusion.”
Maximum Likelihood Training of Score-Based Diffusion Models (2021, methodology): “The other probabilistic model, denoted pODE θ pxq, is derived from the SDE’s associated probability flow ODE [32, 48].”
On Maximum Likelihood Training of Score-Based Generative Models (2021, background): “Every SDE has an associated probability ﬂow ODE whose marginal distribution at each time t matches that SDE [Maoutsa et al., 2020; Song et al., 2021].”
A Survey on Generative Diffusion Models (2022, methodology): “Inspired by Maoutsa et al. [44] and Chen et al. [45], any type of diffusion process can be derived into a special form of ODE.”
Stochastic Interpolants: A Unifying Framework for Flows and Diffusions (2023, background): “Moreover, the dynamics of these stochastic processes also emit a probability flow equation at the level of the distribution, first noted [2, 35, 23] and then applied in [32, 40, 24, 4].”