In this talk, we explore the approximation capabilities of neural networks in various functional spaces, with a particular focus on Fourier-Lebesgue and Barron spaces. These results are based on joint work with Thomas Dittrich.
We begin by extending the classical spectral Barron spaces to anisotropic weighted Fourier-Lebesgue spaces, which incorporate both spatial and temporal domains with different regularities and integrability. This extension allows for the approximation of solutions to evolutionary partial differential equations by shallow networks. We present new results on the approximation error in terms of Bochner-Sobolev norms and derive bounds on the approximation rates for functions in these anisotropic spaces cf. [1].
Further, we delve into weighted Sobolev spaces, extending existing results to functions in spectral Barron spaces over unbounded domains. By considering both bounded domains with Muckenhoupt weights and unbounded domains with decaying weights, we establish asymptotic approximation rates for shallow neural networks that avoid the curse of dimensionality, cf. [2].
Lastly, we briefly address the approximation of operators in infinite-dimensional spaces. We discuss the approximation of Fourier multipliers and the conditions under which neural networks can approximate operator behaviors in Fréchet metrics, cf. [3].
[1] Ahmed Abdeljawad, Thomas Dittrich. Space-Time Approximation with Shallow Neural Networks in Fourier Lebesgue spaces, arXiv:2312.08461
[2] Ahmed Abdeljawad, Thomas Dittrich. Weighted Sobolev Approximation Rates for Neural Networks on Unbounded Domains, arXiv:2411.04108
[3] Ahmed Abdeljawad, Thomas Dittrich. Approximation Rates in Fréchet Metrics: Barron Spaces, Paley-Wiener Spaces, and Fourier Multipliers, arXiv:2501.04023