Sampling Complexity of Deep Approximation Spaces

no code implementations • 20 Dec 2023 • Ahmed Abdeljawad, Philipp Grohs

While it is well-known that neural networks enjoy excellent approximation capabilities, it remains a big challenge to compute such approximations from point samples.

Space-Time Approximation with Shallow Neural Networks in Fourier Lebesgue spaces

no code implementations • 13 Dec 2023 • Ahmed Abdeljawad, Thomas Dittrich

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.

Deep Neural Network Approximation For Hölder Functions

no code implementations • 11 Jan 2022 • Ahmed Abdeljawad

In this work, we explore the approximation capability of deep Rectified Quadratic Unit neural networks for H\"older-regular functions, with respect to the uniform norm.

Integral representations of shallow neural network with Rectified Power Unit activation function

no code implementations • 20 Dec 2021 • Ahmed Abdeljawad, Philipp Grohs

In this effort, we derive a formula for the integral representation of a shallow neural network with the Rectified Power Unit activation function.

Approximations with deep neural networks in Sobolev time-space

no code implementations • 23 Dec 2020 • Ahmed Abdeljawad, Philipp Grohs

Solutions of evolution equation generally lies in certain Bochner-Sobolev spaces, in which the solution may has regularity and integrability properties for the time variable that can be different for the space variables.