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

*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.