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arXiv · 2206.00860

Self-Consistency of the Fokker-Planck Equation

Zebang Shen·Zhenfu Wang·Satyen Kale·Alejandro Ribeiro·Amin Karbasi·Hamed Hassani
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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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Zebang Shen, Zhenfu Wang, Satyen Kale, Alejandro Ribeiro, Amin Karbasi, Hamed Hassani. 2022-06-02. Self-Consistency of the Fokker-Planck Equation. https://arxiv.org/abs/2206.00860

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