Space-Time Approximation with Shallow Neural Networks in Fourier Lebesgue spaces
This work addresses the limitation of static PDE approximations for researchers in numerical analysis and machine learning, though it appears incremental as it builds on existing spectral Barron space theory.
The paper tackles the problem of approximating solutions to time-dependent partial differential equations (PDEs) by extending spectral Barron spaces to anisotropic weighted Fourier-Lebesgue spaces, which allow different regularity in space and time, and establishes a bound on the approximation rate for shallow neural networks in the Bochner-Sobolev norm.
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.