153 citations · 899 across the 45 of their papers we have counts for
12 papers · 1 filter
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…
Deep neural network approximation for high-dimensional elliptic PDEs with boundary conditions
Philipp Grohs, Lukas Herrmann
In recent work it has been established that deep neural networks are capable of approximating solutions to a large class of parabolic partial differential equations without incurri…
Uniform error estimates for artificial neural network approximations for heat equations
Lukas Gonon, Philipp Grohs, Arnulf Jentzen +2
Recently, artificial neural networks (ANNs) in conjunction with stochastic gradient descent optimization methods have been employed to approximately compute solutions of possibly r…
Deep neural network approximations for Monte Carlo algorithms
Philipp Grohs, Arnulf Jentzen, Diyora Salimova
Recently, it has been proposed in the literature to employ deep neural networks (DNNs) together with stochastic gradient descent methods to approximate solutions of PDEs. There are…
Space-time error estimates for deep neural network approximations for differential equations
Philipp Grohs, Fabian Hornung, Arnulf Jentzen +1
Over the last few years deep artificial neural networks (DNNs) have very successfully been used in numerical simulations for a wide variety of computational problems including comp…
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…
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…
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…
Convergence of a Strang splitting finite element discretization for the Schrödinger-Poisson equation
Winfried Auzinger, Thomas Kassebacher, Othmar Koch +1
Operator splitting methods combined with finite element spatial discretizations are studied for time-dependent nonlinear Schrödinger equations. In particular, the Schrödinger-Poiss…
On the Approximation of Functions with Line Singularities by Ridgelets
Axel Obermeier, Philipp Grohs
In [GO15], the authors discussed the existence of numerically feasible solvers for advection equations that run in optimal computational complexity. In this paper, we complete the…
Optimal Adaptive Ridgelet Schemes for Linear Transport Equations
Philipp Grohs, Axel Obermeier
In this paper we present a novel method for the numerical solution of linear transport equations, which is based on ridgelets. Such equations arise for instance in radiative transf…