It is one of the most challenging problems in applied mathematics to approximatively solve
high-dimensional partial differential equations (PDEs). Recently, several deep learning …

We propose a very general framework for deriving rigorous bounds on the approximation
error for physics-informed neural networks (PINNs) and operator learning architectures such …

In recent years, tremendous progress has been made on numerical algorithms for solving
partial differential equations (PDEs) in a very high dimension, using ideas from either …

Physics-informed neural networks approximate solutions of PDEs by minimizing pointwise
residuals. We derive rigorous bounds on the error, incurred by PINNs in approximating the …

We analyze approximation rates by deep ReLU networks of a class of multivariate solutions
of Kolmogorov equations which arise in option pricing. Key technical devices are deep …

Recent years have witnessed a growth in mathematics for deep learning—which seeks a
deeper understanding of the concepts of deep learning with mathematics and explores how …

Recently, several deep learning (DL) methods for approximating high-dimensional partial
differential equations (PDEs) have been proposed. The interest that these methods have …

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 …

Over the last few years deep artificial neural networks (ANNs) have very successfully been
used in numerical simulations for a wide variety of computational problems including …

This paper introduces a new approximation scheme for solving high-dimensional semilinear
partial differential equations (PDEs) and backward stochastic differential equations (BSDEs) …