A semigroup method for high dimensional committor functions based on neural network

Haoya Li, Yuehaw Khoo, Yinuo Ren, Lexing Ying
Proceedings of the 2nd Mathematical and Scientific Machine Learning Conference, PMLR 145:598-618, 2022.

Abstract

This paper proposes a new method based on neural networks for computing the high-dimensional committor functions that satisfy Fokker-Planck equations. Instead of working with partial differ- ential equations, the new method works with an integral formulation based on the semigroup of the differential operator. The variational form of the new formulation is then solved by param- eterizing the committor function as a neural network. There are two major benefits of this new approach. First, stochastic gradient descent type algorithms can be applied in the training of the committor function without the need of computing any mixed second order derivatives. Moreover, unlike the previous methods that enforce the boundary conditions through penalty terms, the new method takes into account the boundary conditions automatically. Numerical results are provided to demonstrate the performance of the proposed method.

Cite this Paper

BibTeX
@InProceedings{pmlr-v145-li22a, title = {A semigroup method for high dimensional committor functions based on neural network}, author = {Li, Haoya and Khoo, Yuehaw and Ren, Yinuo and Ying, Lexing}, booktitle = {Proceedings of the 2nd Mathematical and Scientific Machine Learning Conference}, pages = {598--618}, 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/li22a/li22a.pdf}, url = {https://proceedings.mlr.press/v145/li22a.html}, abstract = { This paper proposes a new method based on neural networks for computing the high-dimensional committor functions that satisfy Fokker-Planck equations. Instead of working with partial differ- ential equations, the new method works with an integral formulation based on the semigroup of the differential operator. The variational form of the new formulation is then solved by param- eterizing the committor function as a neural network. There are two major benefits of this new approach. First, stochastic gradient descent type algorithms can be applied in the training of the committor function without the need of computing any mixed second order derivatives. Moreover, unlike the previous methods that enforce the boundary conditions through penalty terms, the new method takes into account the boundary conditions automatically. Numerical results are provided to demonstrate the performance of the proposed method. } }
Endnote
%0 Conference Paper %T A semigroup method for high dimensional committor functions based on neural network %A Haoya Li %A Yuehaw Khoo %A Yinuo Ren %A Lexing Ying %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-li22a %I PMLR %P 598--618 %U https://proceedings.mlr.press/v145/li22a.html %V 145 %X This paper proposes a new method based on neural networks for computing the high-dimensional committor functions that satisfy Fokker-Planck equations. Instead of working with partial differ- ential equations, the new method works with an integral formulation based on the semigroup of the differential operator. The variational form of the new formulation is then solved by param- eterizing the committor function as a neural network. There are two major benefits of this new approach. First, stochastic gradient descent type algorithms can be applied in the training of the committor function without the need of computing any mixed second order derivatives. Moreover, unlike the previous methods that enforce the boundary conditions through penalty terms, the new method takes into account the boundary conditions automatically. Numerical results are provided to demonstrate the performance of the proposed method.
APA
Li, H., Khoo, Y., Ren, Y. & Ying, L.. (2022). A semigroup method for high dimensional committor functions based on neural network. Proceedings of the 2nd Mathematical and Scientific Machine Learning Conference, in Proceedings of Machine Learning Research 145:598-618 Available from https://proceedings.mlr.press/v145/li22a.html.

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