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

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arxiv 2012.01484 v3 pith:NT4BAOV4 submitted 2020-12-02 math.AP cs.LG

Weinan E , Stephan Wojtowytsch This is my paper
classification math.APcs.LG
keywords spacesbarrondatadifferentialequationsfunctionneuralpartial
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We use explicit representation formulas to show that solutions to certain partial differential equations lie in Barron spaces or multilayer spaces if the PDE data lie in such function spaces. Consequently, 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.

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  1. Elliptic Regularity Theory in Barron Spaces and Applications to the Deep Ritz Method

    math.AP 2026-07 accept novelty 7.0 of 10

    Harmonic functions with Barron Dirichlet data fail to be Lipschitz or H², yet admit Barron approximants of norm ~|log ε| with error ~ε on half-spaces and 2D rectangles, giving Deep Ritz a priori rates.

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