Some observations on high-dimensional partial differential equations with Barron data
This work is significant for researchers in numerical analysis and machine learning, particularly those working on solving high-dimensional PDEs, by showing conditions under which neural network representations are effective.
This paper demonstrates that solutions to certain partial differential equations (PDEs) can be efficiently represented by artificial neural networks, even in high dimensions, provided the PDE data themselves reside in Barron or multilayer spaces. Conversely, it provides examples where solutions do not fit within the neural network's associated function space.
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.