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This article provides a mathematically rigorous introduction to denoising diffusion probabilistic models (DDPMs), sometimes also referred to as diffusion probabilistic models or diffusion models, for generative artificial intelligence. We provide a d
Externí odkaz:
http://arxiv.org/abs/2412.01371
Deep learning methods - consisting of a class of deep neural networks (DNNs) trained by a stochastic gradient descent (SGD) optimization method - are nowadays key tools to solve data driven supervised learning problems. Despite the great success of S
Externí odkaz:
http://arxiv.org/abs/2410.10533
Autor:
Gonon, Lukas, Jentzen, Arnulf, Kuckuck, Benno, Liang, Siyu, Riekert, Adrian, von Wurstemberger, Philippe
The approximation of solutions of partial differential equations (PDEs) with numerical algorithms is a central topic in applied mathematics. For many decades, various types of methods for this purpose have been developed and extensively studied. One
Externí odkaz:
http://arxiv.org/abs/2408.13222
Autor:
Dereich, Steffen, Jentzen, Arnulf
Stochastic gradient descent (SGD) optimization methods are nowadays the method of choice for the training of deep neural networks (DNNs) in artificial intelligence systems. In practically relevant training problems, usually not the plain vanilla stan
Externí odkaz:
http://arxiv.org/abs/2407.21078
Deep learning algorithms - typically consisting of a class of deep neural networks trained by a stochastic gradient descent (SGD) optimization method - are nowadays the key ingredients in many artificial intelligence (AI) systems and have revolutioni
Externí odkaz:
http://arxiv.org/abs/2407.08100
It is known that the standard stochastic gradient descent (SGD) optimization method, as well as accelerated and adaptive SGD optimization methods such as the Adam optimizer fail to converge if the learning rates do not converge to zero (as, for examp
Externí odkaz:
http://arxiv.org/abs/2406.14340
It is a challenging topic in applied mathematics to solve high-dimensional nonlinear partial differential equations (PDEs). Standard approximation methods for nonlinear PDEs suffer under the curse of dimensionality (COD) in the sense that the number
Externí odkaz:
http://arxiv.org/abs/2406.10876
Autor:
Jentzen, Arnulf, Riekert, Adrian
Stochastic gradient descent (SGD) optimization methods such as the plain vanilla SGD method and the popular Adam optimizer are nowadays the method of choice in the training of artificial neural networks (ANNs). Despite the remarkable success of SGD m
Externí odkaz:
http://arxiv.org/abs/2402.05155
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 different artificial neural network (ANN) architectures (such as full
Externí odkaz:
http://arxiv.org/abs/2310.20360
Recently, several deep learning (DL) methods for approximating high-dimensional partial differential equations (PDEs) have been proposed. The interest that these methods have generated in the literature is in large part due to simulations which appea
Externí odkaz:
http://arxiv.org/abs/2309.13722