Scientific machine learning (SciML), an area of research where techniques from machine
learning and scientific computing are combined, has become of increasing importance and …

Physics-informed neural networks (PINNs) encode physical conservation laws and prior
physical knowledge into the neural networks, ensuring the correct physics is represented …

The parareal algorithm is a method to solve time-dependent problems parallel in time: it
approximates parts of the solution later in time simultaneously to parts of the solution earlier …

These notes serve as an introduction to a subject of study in computational mathematics
referred to as domain decomposition methods. It concerns divide and conquer methods for …

We present and analyze a new space-time parallel multigrid method for parabolic equations.
The method is based on arbitrarily high order discontinuous Galerkin discretizations in time …

Physics-Informed Machine Learning (PIML) is an emerging computing paradigm that offers a
new approach to tackle multiphysics modeling problems prevalent in the field of chemical …

After stating an abstract convergence result for the parareal algorithm used in the
parallelization in time of general partial differential equations, we analyze the stability and …

In this article, we motivate, derive, and test effective preconditioners to be used with the
Minres algorithm for solving a number of saddle point systems which arise in PDE …

We discuss the stability of the Parareal algorithm for an autonomous set of differential
equations. The stability function for the algorithm is derived, and stability conditions for the …

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 …