153 citations · 210 across the 6 of their papers we have counts for
9 papers
Space-time deep neural network approximations for high-dimensional partial differential equations
Fabian Hornung, Arnulf Jentzen, Diyora Salimova
It is one of the most challenging issues in applied mathematics to approximately solve high-dimensional partial differential equations (PDEs) and most of the numerical approximatio…
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…
Overcoming the curse of dimensionality in the numerical approximation of Allen-Cahn partial differential equations via truncated full-history recursive multilevel Picard approximations
Christian Beck, Fabian Hornung, Martin Hutzenthaler +2
One of the most challenging problems in applied mathematics is the approximate solution of nonlinear partial differential equations (PDEs) in high dimensions. Standard deterministi…
The stochastic nonlinear Schrödinger equation in unbounded domains and manifolds
Fabian Hornung
In this article, we construct a global martingale solution to a general nonlinear Schrödinger equation with linear multiplicative noise in the Stratonovich form. Our framework incl…
Weak martingale solutions for the stochastic nonlinear Schrödinger equation driven by pure jump noise
Zdzisław Brzeźniak, Fabian Hornung, Utpal Manna
We construct a martingale solution of the stochastic nonlinear Schrödinger equation with a multiplicative noise of jump type in the Marcus canonical form. The problem is formulated…
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…
Uniqueness of martingale solutions for the stochastic nonlinear Schrödinger equation on 3d compact manifolds
Zdzislaw Brzezniak, Fabian Hornung, Lutz Weis
We prove pathwise uniqueness for solutions of the nonlinear Schrödinger equation with conservative multiplicative noise on compact 3D manifolds. In particular, we generalize the re…
Martingale solutions for the stochastic nonlinear Schrödinger equation in the energy space
Zdzislaw Brzezniak, Fabian Hornung, Lutz Weis
We consider a stochastic nonlinear Schrödinger equation with multiplicative noise in an abstract framework that covers subcritical focusing and defocusing stochastic NLS in o…
The nonlinear stochastic Schrödinger equation via stochastic Strichartz estimates
Fabian Hornung
We consider the stochastic NLS with nonlinear Stratonovic noise for initial values in and prove local existence and uniqueness of a mild solution for subcritical and cri…