153 citations · 899 across the 45 of their papers we have counts for
7 papers · 1 filter
DNN Expression Rate Analysis of High-dimensional PDEs: Application to Option Pricing
Dennis Elbrächter, Philipp Grohs, Arnulf Jentzen +1
We analyze approximation rates by deep ReLU networks of a class of multi-variate solutions of Kolmogorov equations which arise in option pricing. Key technical devices are deep ReL…
Analysis of the Generalization Error: Empirical Risk Minimization over Deep Artificial Neural Networks Overcomes the Curse of Dimensionality in the Numerical Approximation of Black-Scholes Partial Differential Equations
Julius Berner, Philipp Grohs, Arnulf Jentzen
The development of new classification and regression algorithms based on empirical risk minimization (ERM) over deep neural network hypothesis classes, coined deep learning, revolu…
A proof that artificial neural networks overcome the curse of dimensionality in the numerical approximation of Black-Scholes partial differential equations
Philipp Grohs, Fabian Hornung, Arnulf Jentzen +1
Artificial neural networks (ANNs) have very successfully been used in numerical simulations for a series of computational problems ranging from image classification/image recogniti…
The universal approximation power of finite-width deep ReLU networks
Dmytro Perekrestenko, Philipp Grohs, Dennis Elbrächter +1
We show that finite-width deep ReLU neural networks yield rate-distortion optimal approximation (Bölcskei et al., 2018) of polynomials, windowed sinusoidal functions, one-dimension…
Solving the Kolmogorov PDE by means of deep learning
Christian Beck, Sebastian Becker, Philipp Grohs +2
Stochastic differential equations (SDEs) and the Kolmogorov partial differential equations (PDEs) associated to them have been widely used in models from engineering, finance, and…
Gabor phase retrieval is severely ill-posed
Rima Alaifari, Philipp Grohs
The problem of reconstructing a function from the magnitudes of its frame coefficients has recently been shown to be never uniformly stable in infinite-dimensional spaces [5]. This…
Projection-Based Finite Elements for Nonlinear Function Spaces
Philipp Grohs, Hanne Hardering, Oliver Sander +1
We introduce a novel type of approximation spaces for functions with values in a nonlinear manifold. The discrete functions are constructed by piecewise polynomial interpolation in…