Some observations on high-dimensional partial differential equations with Barron data

Weinan E, Stephan Wojtowytsch
Proceedings of the 2nd Mathematical and Scientific Machine Learning Conference, PMLR 145:253-269, 2022.

Abstract

We use explicit representation formulas to show that solutions to certain partial differential equa- tions lie in Barron spaces or multilayer spaces if the PDE data lie in such function spaces. Conse- quently, these solutions can be represented efficiently using artificial neural networks, even in high dimension. Conversely, we present examples in which the solution fails to lie in the function space associated to a neural network under consideration.

Cite this Paper

BibTeX
@InProceedings{pmlr-v145-e22a, title = {Some observations on high-dimensional partial differential equations with Barron data}, author = {E, Weinan and Wojtowytsch, Stephan}, booktitle = {Proceedings of the 2nd Mathematical and Scientific Machine Learning Conference}, pages = {253--269}, 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/e22a/e22a.pdf}, url = {https://proceedings.mlr.press/v145/e22a.html}, abstract = { We use explicit representation formulas to show that solutions to certain partial differential equa- tions lie in Barron spaces or multilayer spaces if the PDE data lie in such function spaces. Conse- quently, these solutions can be represented efficiently using artificial neural networks, even in high dimension. Conversely, we present examples in which the solution fails to lie in the function space associated to a neural network under consideration.} }
Endnote
%0 Conference Paper %T Some observations on high-dimensional partial differential equations with Barron data %A Weinan E %A Stephan Wojtowytsch %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-e22a %I PMLR %P 253--269 %U https://proceedings.mlr.press/v145/e22a.html %V 145 %X We use explicit representation formulas to show that solutions to certain partial differential equa- tions lie in Barron spaces or multilayer spaces if the PDE data lie in such function spaces. Conse- quently, these solutions can be represented efficiently using artificial neural networks, even in high dimension. Conversely, we present examples in which the solution fails to lie in the function space associated to a neural network under consideration.
APA
E, W. & Wojtowytsch, S.. (2022). Some observations on high-dimensional partial differential equations with Barron data. Proceedings of the 2nd Mathematical and Scientific Machine Learning Conference, in Proceedings of Machine Learning Research 145:253-269 Available from https://proceedings.mlr.press/v145/e22a.html.

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