153 citations · 899 across the 45 of their papers we have counts for
10 papers · 1 filter
Integral representations of shallow neural network with Rectified Power Unit activation function
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. Mainly, our first result deals wi…
On foundational discretization barriers in STFT phase retrieval
Philipp Grohs, Lukas Liehr
We prove that there exists no window function and no lattice such that every is determined up to…
Sobolev-type embeddings for neural network approximation spaces
Philipp Grohs, Felix Voigtlaender
We consider neural network approximation spaces that classify functions according to the rate at which they can be approximated (with error measured in ) by ReLU neural networ…
Stable Gabor phase retrieval in Gaussian shift-invariant spaces via biorthogonality
Philipp Grohs, Lukas Liehr
We study the phase reconstruction of signals belonging to complex Gaussian shift-invariant spaces from spectrogram measurements where $\mathc…
-stability analysis for Gabor phase retrieval
Philipp Grohs, Martin Rathmair
We consider the problem of reconstructing the missing phase information from spectrogram data with $$ \mathcal{G}f(x,y)=\int_\mathbb{R} f(t) e^{-π(t-x)^2}e^{-2πi…
Solving the electronic Schrödinger equation for multiple nuclear geometries with weight-sharing deep neural networks
Michael Scherbela, Rafael Reisenhofer, Leon Gerard +2
Accurate numerical solutions for the Schrödinger equation are of utmost importance in quantum chemistry. However, the computational cost of current high-accuracy methods scales poo…
The Modern Mathematics of Deep Learning
Julius Berner, Philipp Grohs, Gitta Kutyniok +1
We describe the new field of mathematical analysis of deep learning. This field emerged around a list of research questions that were not answered within the classical framework of…
Proof of the Theory-to-Practice Gap in Deep Learning via Sampling Complexity bounds for Neural Network Approximation Spaces
Philipp Grohs, Felix Voigtlaender
We study the computational complexity of (deterministic or randomized) algorithms based on point samples for approximating or integrating functions that can be well approximated by…
Deep neural network approximation for high-dimensional parabolic Hamilton-Jacobi-Bellman equations
Philipp Grohs, Lukas Herrmann
The approximation of solutions to second order Hamilton--Jacobi--Bellman (HJB) equations by deep neural networks is investigated. It is shown that for HJB equations that arise in t…
Lower bounds for artificial neural network approximations: A proof that shallow neural networks fail to overcome the curse of dimensionality
Philipp Grohs, Shokhrukh Ibragimov, Arnulf Jentzen +1
Artificial neural networks (ANNs) have become a very powerful tool in the approximation of high-dimensional functions. Especially, deep ANNs, consisting of a large number of hidden…