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06/29/2023
Efficient Sobolev approximation of linear parabolic PDEs in high dimensions
In this paper, we study the error in first order Sobolev norm in the app...
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03/24/2023
Optimal transport and Wasserstein distances for causal models
In this paper we introduce a variant of optimal transport adapted to the...
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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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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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12/03/2021
Computation of conditional expectations with guarantees
Theoretically, the conditional expectation of a square-integrable random...
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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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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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12/08/2020
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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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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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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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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07/08/2019
