6 papers
Adam symmetry theorem: characterization of the convergence of the stochastic Adam optimizer
Steffen Dereich, Thang Do, Arnulf Jentzen +1
Beside the standard stochastic gradient descent (SGD) method, the Adam optimizer due to Kingma & Ba (2014) is currently probably the best-known optimization method for the training…
Physics-informed diffusion models in spectral space
Davide Gallon, Philippe von Wurstemberger, Patrick Cheridito +1
We propose physics-informed spectral diffusion (PISD), a methodology that combines generative latent diffusion models with physics-informed machine learning to generate solutions o…
Algorithmically Designed Artificial Neural Networks (ADANNs): Higher order deep operator learning for parametric partial differential equations
Arnulf Jentzen, Adrian Riekert, Philippe von Wurstemberger
In this article we propose a new deep learning approach to approximate operators related to parametric partial differential equations (PDEs). In particular, we introduce a new stra…
Mathematical Introduction to Deep Learning: Methods, Implementations, and Theory
Arnulf Jentzen, Benno Kuckuck, Philippe von Wurstemberger
This book aims to provide an introduction to the topic of deep learning algorithms. We review essential components of deep learning algorithms in full mathematical detail including…
High-dimensional approximation spaces of artificial neural networks and applications to partial differential equations
Pierfrancesco Beneventano, Patrick Cheridito, Arnulf Jentzen +1
In this paper we develop a new machinery to study the capacity of artificial neural networks (ANNs) to approximate high-dimensional functions without suffering from the curse of di…
An overview of diffusion models for generative artificial intelligence
Davide Gallon, Arnulf Jentzen, Philippe von Wurstemberger
This article provides a mathematically rigorous introduction to denoising diffusion probabilistic models (DDPMs), sometimes also referred to as diffusion probabilistic models or di…