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We use explicit representation formulas to show that solutions to certainpartial differential equations lie in Barron spaces or multilayer spaces if thePDE data lie in such function spaces. Consequently, these solutions can berepresented efficiently using artificial neural networks, even in highdimension. Conversely, we present examples in which the solution fails to liein the function space associated to a neural network under consideration.

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