A deep learning method for solving Fokker-Planck equations

Jiayu Zhai, Matthew Dobson, Yao Li
Proceedings of the 2nd Mathematical and Scientific Machine Learning Conference, PMLR 145:568-597, 2022.

Abstract

The time evolution of the probability distribution of a stochastic differential equation follows the Fokker-Planck equation, which usually has an unbounded, high-dimensional domain. Inspired by Li (2019), we propose a mesh-free Fokker-Planck solver, in which the solution to the Fokker-Planck equation is now represented by a neural network. The presence of the differential operator in the loss function improves the accuracy of the neural network representation and reduces the demand of data in the training process. Several high dimensional numerical examples are demonstrated.

Cite this Paper

BibTeX
@InProceedings{pmlr-v145-zhai22a, title = {A deep learning method for solving Fokker-Planck equations}, author = {Zhai, Jiayu and Dobson, Matthew and Li, Yao}, booktitle = {Proceedings of the 2nd Mathematical and Scientific Machine Learning Conference}, pages = {568--597}, year = {2022}, editor = {Bruna, Joan and Hesthaven, Jan and Zdeborova, Lenka}, volume = {145}, series = {Proceedings of Machine Learning Research}, month = {16--19 Aug}, publisher = {PMLR}, pdf = {https://proceedings.mlr.press/v145/zhai22a/zhai22a.pdf}, url = {https://proceedings.mlr.press/v145/zhai22a.html}, abstract = {The time evolution of the probability distribution of a stochastic differential equation follows the Fokker-Planck equation, which usually has an unbounded, high-dimensional domain. Inspired by Li (2019), we propose a mesh-free Fokker-Planck solver, in which the solution to the Fokker-Planck equation is now represented by a neural network. The presence of the differential operator in the loss function improves the accuracy of the neural network representation and reduces the demand of data in the training process. Several high dimensional numerical examples are demonstrated. } }
Endnote
%0 Conference Paper %T A deep learning method for solving Fokker-Planck equations %A Jiayu Zhai %A Matthew Dobson %A Yao Li %B Proceedings of the 2nd Mathematical and Scientific Machine Learning Conference %C Proceedings of Machine Learning Research %D 2022 %E Joan Bruna %E Jan Hesthaven %E Lenka Zdeborova %F pmlr-v145-zhai22a %I PMLR %P 568--597 %U https://proceedings.mlr.press/v145/zhai22a.html %V 145 %X The time evolution of the probability distribution of a stochastic differential equation follows the Fokker-Planck equation, which usually has an unbounded, high-dimensional domain. Inspired by Li (2019), we propose a mesh-free Fokker-Planck solver, in which the solution to the Fokker-Planck equation is now represented by a neural network. The presence of the differential operator in the loss function improves the accuracy of the neural network representation and reduces the demand of data in the training process. Several high dimensional numerical examples are demonstrated.
APA
Zhai, J., Dobson, M. & Li, Y.. (2022). A deep learning method for solving Fokker-Planck equations. Proceedings of the 2nd Mathematical and Scientific Machine Learning Conference, in Proceedings of Machine Learning Research 145:568-597 Available from https://proceedings.mlr.press/v145/zhai22a.html.

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