Space-Time Approximation with Shallow Neural Networks in Fourier
  Lebesgue spaces
Ahmed Abdeljawad Thomas Dittrich
Approximation capabilities of shallow neural networks (SNNs) form an integral
part in understanding the properties of deep neural networks (DNNs). In the
study of these approximation capabilities some very popular classes of target
functions are the so-called spectral Barron spaces. This spaces are of special
interest when it comes to the approximation of partial differential equation
(PDE) solutions. It has been shown that the solution of certain static PDEs
will lie in some spectral Barron space. In order to alleviate the limitation to
static PDEs and include a time-domain that might have a different regularity
than the space domain, we extend the notion of spectral Barron spaces to
anisotropic weighted Fourier-Lebesgue spaces. In doing so, we consider target
functions that have two blocks of variables, among which each block is allowed
to have different decay and integrability properties. For these target
functions we first study the inclusion of anisotropic weighted Fourier-Lebesgue
spaces in the Bochner-Sobolev spaces. With that we can now also measure the
approximation error in terms of an anisotropic Sobolev norm, namely the
Bochner-Sobolev norm. We use this observation in a second step where we
establish a bound on the approximation rate for functions from the anisotropic
weighted Fourier-Lebesgue spaces and approximation via SNNs in the
Bochner-Sobolev norm.