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Deep neural network approximation theory for high-dimensional functions
Pierfrancesco Beneventano, Patrick Cheridito, Robin Graeber +2
The purpose of this article is to develop a machinery to study the capacity of deep neural networks (DNNs) to approximate high-dimensional functions. In particular, we show that DN…
On the existence of global minima and convergence analyses for gradient descent methods in the training of deep neural networks
Arnulf Jentzen, Adrian Riekert
In this article we study fully-connected feedforward deep ReLU ANNs with an arbitrarily large number of hidden layers and we prove convergence of the risk of the GD optimization me…
Convergence proof for stochastic gradient descent in the training of deep neural networks with ReLU activation for constant target functions
Martin Hutzenthaler, Arnulf Jentzen, Katharina Pohl +2
In many numerical simulations stochastic gradient descent (SGD) type optimization methods perform very effectively in the training of deep neural networks (DNNs) but till this day…
Strong -error analysis of nonlinear Monte Carlo approximations for high-dimensional semilinear partial differential equations
Martin Hutzenthaler, Arnulf Jentzen, Benno Kuckuck +1
Full-history recursive multilevel Picard (MLP) approximation schemes have been shown to overcome the curse of dimensionality in the numerical approximation of high-dimensional semi…
Overcoming the curse of dimensionality in the numerical approximation of backward stochastic differential equations
Martin Hutzenthaler, Arnulf Jentzen, Thomas Kruse +1
Backward stochastic differential equations (BSDEs) belong nowadays to the most frequently studied equations in stochastic analysis and computational stochastics. BSDEs in applicati…
Existence, uniqueness, and convergence rates for gradient flows in the training of artificial neural networks with ReLU activation
Simon Eberle, Arnulf Jentzen, Adrian Riekert +1
The training of artificial neural networks (ANNs) with rectified linear unit (ReLU) activation via gradient descent (GD) type optimization schemes is nowadays a common industrially…