An overview on deep learning-based approximation methods for partial differential equations
It is one of the most challenging problems in applied mathematics to approximatively solve
high-dimensional partial differential equations (PDEs). Recently, several deep learning …
high-dimensional partial differential equations (PDEs). Recently, several deep learning …
An extreme learning machine-based method for computational PDEs in higher dimensions
We present two effective methods for solving high-dimensional partial differential equations
(PDE) based on randomized neural networks. Motivated by the universal approximation …
(PDE) based on randomized neural networks. Motivated by the universal approximation …
Approximation bounds for random neural networks and reservoir systems
This work studies approximation based on single-hidden-layer feedforward and recurrent
neural networks with randomly generated internal weights. These methods, in which only …
neural networks with randomly generated internal weights. These methods, in which only …
Universal approximation theorem and error bounds for quantum neural networks and quantum reservoirs
Universal approximation theorems are the foundations of classical neural networks,
providing theoretical guarantees that the latter are able to approximate maps of interest …
providing theoretical guarantees that the latter are able to approximate maps of interest …
Deep ReLU neural networks overcome the curse of dimensionality for partial integrodifferential equations
Deep neural networks (DNNs) with ReLU activation function are proved to be able to
express viscosity solutions of linear partial integrodifferential equations (PIDEs) on state …
express viscosity solutions of linear partial integrodifferential equations (PIDEs) on state …
Infinite-dimensional reservoir computing
Reservoir computing approximation and generalization bounds are proved for a new
concept class of input/output systems that extends the so-called generalized Barron …
concept class of input/output systems that extends the so-called generalized Barron …
Mathematical introduction to deep learning: methods, implementations, and theory
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 …
essential components of deep learning algorithms in full mathematical detail including …
Chefs' random tables: Non-trigonometric random features
We introduce chefs' random tables (CRTs), a new class of non-trigonometric random
features (RFs) to approximate Gaussian and softmax kernels. CRTs are an alternative to …
features (RFs) to approximate Gaussian and softmax kernels. CRTs are an alternative to …
Dense-exponential random features: sharp positive estimators of the Gaussian kernel
The problem of efficient approximation of a linear operator induced by the Gaussian or
softmax kernel is often addressed using random features (RFs) which yield an unbiased …
softmax kernel is often addressed using random features (RFs) which yield an unbiased …
[PDF][PDF] Universal approximation property of random neural networks
ARIEL NEUFELD AND PHILIPP SCHMOCKER ABSTRACT. In this paper, we study random
neural networks which are single-hidden-layer feedforward neural networks whose weights …
neural networks which are single-hidden-layer feedforward neural networks whose weights …