Self-Consistency of the Fokker Planck Equation

Zebang Shen, Zhenfu Wang, Satyen Kale, Alejandro Ribeiro, Amin Karbasi, Hamed Hassani
Proceedings of Thirty Fifth Conference on Learning Theory, PMLR 178:817-841, 2022.

Abstract

The Fokker-Planck equation (FPE) is the partial differential equation that governs the density evolution of the Ito 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.

Cite this Paper

BibTeX
@InProceedings{pmlr-v178-shen22a, title = {Self-Consistency of the Fokker Planck Equation}, author = {Shen, Zebang and Wang, Zhenfu and Kale, Satyen and Ribeiro, Alejandro and Karbasi, Amin and Hassani, Hamed}, booktitle = {Proceedings of Thirty Fifth Conference on Learning Theory}, pages = {817--841}, year = {2022}, editor = {Loh, Po-Ling and Raginsky, Maxim}, volume = {178}, series = {Proceedings of Machine Learning Research}, month = {02--05 Jul}, publisher = {PMLR}, pdf = {https://proceedings.mlr.press/v178/shen22a/shen22a.pdf}, url = {https://proceedings.mlr.press/v178/shen22a.html}, abstract = {The Fokker-Planck equation (FPE) is the partial differential equation that governs the density evolution of the Ito 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.} }
Endnote
%0 Conference Paper %T Self-Consistency of the Fokker Planck Equation %A Zebang Shen %A Zhenfu Wang %A Satyen Kale %A Alejandro Ribeiro %A Amin Karbasi %A Hamed Hassani %B Proceedings of Thirty Fifth Conference on Learning Theory %C Proceedings of Machine Learning Research %D 2022 %E Po-Ling Loh %E Maxim Raginsky %F pmlr-v178-shen22a %I PMLR %P 817--841 %U https://proceedings.mlr.press/v178/shen22a.html %V 178 %X The Fokker-Planck equation (FPE) is the partial differential equation that governs the density evolution of the Ito 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.
APA
Shen, Z., Wang, Z., Kale, S., Ribeiro, A., Karbasi, A. & Hassani, H.. (2022). Self-Consistency of the Fokker Planck Equation. Proceedings of Thirty Fifth Conference on Learning Theory, in Proceedings of Machine Learning Research 178:817-841 Available from https://proceedings.mlr.press/v178/shen22a.html.

Related Material