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02/28/2023
On the existence of minimizers in shallow residual ReLU neural network optimization landscapes
Many mathematical convergence results for gradient descent (GD) based al...
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02/07/2023
Algorithmically Designed Artificial Neural Networks (ADANNs): Higher order deep operator learning for parametric partial differential equations
In this article we propose a new deep learning approach to solve paramet...
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01/19/2023
The necessity of depth for artificial neural networks to approximate certain classes of smooth and bounded functions without the curse of dimensionality
In this article we study high-dimensional approximation capacities of sh...
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10/24/2022
An efficient Monte Carlo scheme for Zakai equations
In this paper we develop a numerical method for efficiently approximatin...
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08/03/2022
Gradient descent provably escapes saddle points in the training of shallow ReLU networks
Dynamical systems theory has recently been applied in optimization to pr...
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07/13/2022
Normalized gradient flow optimization in the training of ReLU artificial neural networks
The training of artificial neural networks (ANNs) is nowadays a highly r...
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06/27/2022
On bounds for norms of reparameterized ReLU artificial neural network parameters: sums of fractional powers of the Lipschitz norm control the network parameter vector
It is an elementary fact in the scientific literature that the Lipschitz...
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05/07/2022
Deep learning approximations for non-local nonlinear PDEs with Neumann boundary conditions
Nonlinear partial differential equations (PDEs) are used to model dynami...
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02/06/2022
Learning the random variables in Monte Carlo simulations with stochastic gradient descent: Machine learning for parametric PDEs and financial derivative pricing
In financial engineering, prices of financial products are computed appr...
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12/29/2021
Deep neural network approximation theory for high-dimensional functions
The purpose of this article is to develop machinery to study the capacit...
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12/17/2021
On the existence of global minima and convergence analyses for gradient descent methods in the training of deep neural networks
In this article we study fully-connected feedforward deep ReLU ANNs with...
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12/13/2021
Convergence proof for stochastic gradient descent in the training of deep neural networks with ReLU activation for constant target functions
In many numerical simulations stochastic gradient descent (SGD) type opt...
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10/15/2021
Strong L^p-error analysis of nonlinear Monte Carlo approximations for high-dimensional semilinear partial differential equations
Full-history recursive multilevel Picard (MLP) approximation schemes hav...
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08/24/2021
Overcoming the curse of dimensionality in the numerical approximation of backward stochastic differential equations
Backward stochastic differential equations (BSDEs) belong nowadays to th...
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08/18/2021
Existence, uniqueness, and convergence rates for gradient flows in the training of artificial neural networks with ReLU activation
The training of artificial neural networks (ANNs) with rectified linear ...
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08/10/2021
A proof of convergence for the gradient descent optimization method with random initializations in the training of neural networks with ReLU activation for piecewise linear tar
Gradient descent (GD) type optimization methods are the standard instrum...
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07/09/2021
Convergence analysis for gradient flows in the training of artificial neural networks with ReLU activation
Gradient descent (GD) type optimization schemes are the standard methods...
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04/01/2021
A proof of convergence for stochastic gradient descent in the training of artificial neural networks with ReLU activation for constant target functions
In this article we study the stochastic gradient descent (SGD) optimizat...
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03/19/2021
Landscape analysis for shallow ReLU neural networks: complete classification of critical points for affine target functions
In this paper, we analyze the landscape of the true loss of a ReLU neura...
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03/07/2021
Lower bounds for artificial neural network approximations: A proof that shallow neural networks fail to overcome the curse of dimensionality
Artificial neural networks (ANNs) have become a very powerful tool in th...
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03/03/2021
Full history recursive multilevel Picard approximations for ordinary differential equations with expectations
We consider ordinary differential equations (ODEs) which involve expecta...
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02/23/2021
Convergence rates for gradient descent in the training of overparameterized artificial neural networks with biases
In recent years, artificial neural networks have developed into a powerf...
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02/19/2021
A proof of convergence for gradient descent in the training of artificial neural networks for constant target functions
Gradient descent optimization algorithms are the standard ingredients th...
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12/22/2020
An overview on deep learning-based approximation methods for partial differential equations
It is one of the most challenging problems in applied mathematics to app...
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12/15/2020
Strong overall error analysis for the training of artificial neural networks via random initializations
Although deep learning based approximation algorithms have been applied ...
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12/08/2020
High-dimensional approximation spaces of artificial neural networks and applications to partial differential equations
In this paper we develop a new machinery to study the capacity of artifi...
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12/02/2020
Deep learning based numerical approximation algorithms for stochastic partial differential equations and high-dimensional nonlinear filtering problems
In this article we introduce and study a deep learning based approximati...
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09/29/2020
Nonlinear Monte Carlo methods with polynomial runtime for high-dimensional iterated nested expectations
The approximative calculation of iterated nested expectations is a recur...
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09/05/2020
Multilevel Picard approximations for high-dimensional semilinear second-order PDEs with Lipschitz nonlinearities
The recently introduced full-history recursive multilevel Picard (MLP) a...
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08/31/2020
Algorithms for Solving High Dimensional PDEs: From Nonlinear Monte Carlo to Machine Learning
In recent years, tremendous progress has been made on numerical algorith...
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07/03/2020
Weak error analysis for stochastic gradient descent optimization algorithms
Stochastic gradient descent (SGD) type optimization schemes are fundamen...
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06/12/2020
Non-convergence of stochastic gradient descent in the training of deep neural networks
Deep neural networks have successfully been trained in various applicati...
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06/03/2020
Space-time deep neural network approximations for high-dimensional partial differential equations
It is one of the most challenging issues in applied mathematics to appro...
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05/20/2020
Numerical simulations for full history recursive multilevel Picard approximations for systems of high-dimensional partial differential equations
One of the most challenging issues in applied mathematics is to develop ...
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03/03/2020
Overall error analysis for the training of deep neural networks via stochastic gradient descent with random initialisation
In spite of the accomplishments of deep learning based algorithms in num...
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03/01/2020
Overcoming the curse of dimensionality in the numerical approximation of high-dimensional semilinear elliptic partial differential equations
Recently, so-called full-history recursive multilevel Picard (MLP) appro...
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12/09/2019
Efficient approximation of high-dimensional functions with deep neural networks
In this paper, we develop an approximation theory for deep neural networ...
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12/05/2019
Overcoming the curse of dimensionality in the numerical approximation of parabolic partial differential equations with gradient-dependent nonlinearities
Partial differential equations (PDEs) are a fundamental tool in the mode...
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11/20/2019
Uniform error estimates for artificial neural network approximations for heat equations
Recently, artificial neural networks (ANNs) in conjunction with stochast...
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11/08/2019
Generalised multilevel Picard approximations
It is one of the most challenging problems in applied mathematics to app...
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11/04/2019
Strong convergence rates on the whole probability space for space-time discrete numerical approximation schemes for stochastic Burgers equations
The main result of this article establishes strong convergence rates on ...
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09/30/2019
Full error analysis for the training of deep neural networks
Deep learning algorithms have been applied very successfully in recent y...
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08/28/2019
Deep neural network approximations for Monte Carlo algorithms
Recently, it has been proposed in the literature to employ deep neural n...
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08/11/2019
Space-time error estimates for deep neural network approximations for differential equations
Over the last few years deep artificial neural networks (DNNs) have very...
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08/05/2019
Solving high-dimensional optimal stopping problems using deep learning
Nowadays many financial derivatives which are traded on stock and future...
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07/15/2019
Overcoming the curse of dimensionality in the numerical approximation of Allen-Cahn partial differential equations via truncated full-history recursive multilevel Picard approx
One of the most challenging problems in applied mathematics is the appro...
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07/08/2019
Deep splitting method for parabolic PDEs
In this paper we introduce a numerical method for parabolic PDEs that co...
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05/13/2019
Towards a regularity theory for ReLU networks -- chain rule and global error estimates
Although for neural networks with locally Lipschitz continuous activatio...
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04/02/2019
Convergence rates for the stochastic gradient descent method for non-convex objective functions
We prove the local convergence to minima and estimates on the rate of co...
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09/09/2018