Some observations on partial differential equations in Barron and multilayer spaces
Abstract
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.
 Publication:

arXiv eprints
 Pub Date:
 December 2020
 arXiv:
 arXiv:2012.01484
 Bibcode:
 2020arXiv201201484E
 Keywords:

 Mathematics  Analysis of PDEs;
 Computer Science  Machine Learning;
 68T07;
 35C15;
 65M80