Probability flow solution of the Fokker-Planck equation
- URL: http://arxiv.org/abs/2206.04642v1
- Date: Thu, 9 Jun 2022 17:37:09 GMT
- Title: Probability flow solution of the Fokker-Planck equation
- Authors: Nicholas M. Boffi and Eric Vanden-Eijnden
- Abstract summary: We introduce an alternative scheme based on integrating an ordinary differential equation that describes the flow of probability.
Unlike the dynamics, this equation deterministically pushes samples from the initial density onto samples from the solution at any later time.
Our approach is based on recent advances in score-based diffusion for generative modeling.
- Score: 10.484851004093919
- License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
- Abstract: The method of choice for integrating the time-dependent Fokker-Planck
equation in high-dimension is to generate samples from the solution via
integration of the associated stochastic differential equation. Here, we
introduce an alternative scheme based on integrating an ordinary differential
equation that describes the flow of probability. Unlike the stochastic
dynamics, this equation deterministically pushes samples from the initial
density onto samples from the solution at any later time. The method has the
advantage of giving direct access to quantities that are challenging to
estimate only given samples from the solution, 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 particles according
to the instantaneous probability current. Our approach is based on recent
advances in score-based diffusion for generative modeling, with the important
difference that the training procedure is self-contained and does not require
samples from the target density to be available beforehand. To demonstrate the
validity of the approach, we consider several examples from the physics of
interacting particle systems; we find that the method scales well to
high-dimensional systems, and accurately matches available analytical solutions
and moments computed via Monte-Carlo.
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