Self-Consistency of the Fokker-Planck Equation

• URL: http://arxiv.org/abs/2206.00860v1
• Date: Thu, 2 Jun 2022 03:44:23 GMT
• Title: Self-Consistency of the Fokker-Planck Equation
• Authors: Zebang Shen, Zhenfu Wang, Satyen Kale, Alejandro Ribeiro, Aim Karbasi, Hamed Hassani
• Abstract summary: The Fokker-Planck equation governs the density evolution of the Ito process. Ground-truth velocity field can be shown to be the solution of a fixed-point equation. In this paper, we exploit this concept to design a potential function of the hypothesis velocity fields.
• Score: 117.17004717792344
• License: http://creativecommons.org/licenses/by/4.0/
• Abstract: The Fokker-Planck equation (FPE) is the partial differential equation that governs the density evolution of the It\^o process and is of great importance to the literature of statistical physics and machine learning. The FPE can be regarded as a continuity equation where the change of the density is completely determined by a time varying velocity field. Importantly, this velocity field also depends on the current density function. As a result, the ground-truth velocity field can be shown to be the solution of a fixed-point equation, a property that we call self-consistency. In this paper, we exploit this concept to design a potential function of the hypothesis velocity fields, and prove that, if such a function diminishes to zero during the training procedure, the trajectory of the densities generated by the hypothesis velocity fields converges to the solution of the FPE in the Wasserstein-2 sense. The proposed potential function is amenable to neural-network based parameterization as the stochastic gradient with respect to the parameter can be efficiently computed. Once a parameterized model, such as Neural Ordinary Differential Equation is trained, we can generate the entire trajectory to the FPE.

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