48 citations · 151 across the 14 of their papers we have counts for
8 papers · 1 filter
Residual neural networks overcome the curse of dimensionality for semilinear heat equations
Ilkhom Mukhammadiev, Diyora Salimova
Rigorous results show that feedforward neural networks can overcome the curse of dimensionality in the numerical approximation of high-dimensional partial differential equations (P…
Approximation properties of Residual Neural Networks for Kolmogorov PDEs
Jonas Baggenstos, Diyora Salimova
In recent years residual neural networks (ResNets) as introduced by [He, K., Zhang, X., Ren, S., and Sun, J., Proceedings of the IEEE conference on computer vision and pattern reco…
Weak error analysis for stochastic gradient descent optimization algorithms
Aritz Bercher, Lukas Gonon, Arnulf Jentzen +1
Stochastic gradient descent (SGD) type optimization schemes are fundamental ingredients in a large number of machine learning based algorithms. In particular, SGD type optimization…
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…
Strong and weak divergence of exponential and linear-implicit Euler approximations for stochastic partial differential equations with superlinearly growing nonlinearities
Matteo Beccari, Martin Hutzenthaler, Arnulf Jentzen +3
The explicit Euler scheme and similar explicit approximation schemes (such as the Milstein scheme) are known to diverge strongly and numerically weakly in the case of one-dimension…
A proof that deep artificial neural networks overcome the curse of dimensionality in the numerical approximation of Kolmogorov partial differential equations with constant diffusion and nonlinear drift coefficients
Arnulf Jentzen, Diyora Salimova, Timo Welti
In recent years deep artificial neural networks (DNNs) have been successfully employed in numerical simulations for a multitude of computational problems including, for example, ob…
Strong convergence for explicit space-time discrete numerical approximation methods for stochastic Burgers equations
Arnulf Jentzen, Diyora Salimova, Timo Welti
In this paper we propose and analyze explicit space-time discrete numerical approximations for additive space-time white noise driven stochastic partial differential equations (SPD…
On stochastic differential equations with arbitrarily slow convergence rates for strong approximation in two space dimensions
Máté Gerencsér, Arnulf Jentzen, Diyora Salimova
In the recent article [Jentzen, A., Müller-Gronbach, T., and Yaroslavtseva, L., Commun. Math. Sci., 14(6), 1477--1500, 2016] it has been established that for every arbitrarily slow…