The objective of this book is to introduce the reader to the Localized Orthogonal
Decomposition (LOD) method for solving partial differential equations with multiscale data …

Physics-informed neural network (PINN) is a data-driven approach to solving equations. It is
successful in many applications; however, the accuracy of the PINN is not satisfactory when …

In this paper, we propose a rigorous and accurate non-local (in the oversampled region)
upscaling framework based on some recently developed multiscale methods [10]. Our …

Operator learning trains a neural network to map functions to functions. An ideal operator
learning framework should be mesh-free in the sense that the training does not require a …

The objective of this paper is to design novel multi-layer neural networks for multiscale
simulations of flows taking into account the observed fine data and physical modeling …

We provide a concise review of the exponentially convergent multiscale finite element
method (ExpMsFEM) for efficient model reduction of PDEs in heterogeneous media without …

In this paper, a deep learning method using Koopman operator is presented for modeling
nonlinear multiscale dynamical problems. Koopman operator is able to transform a non …

In this paper, we present a general derivation of multicontinuum equations and discuss cell
problems. We present constraint cell problem formulations in a representative volume …

Inherently coupled flow and geomechanics processes in fractured shale media have
implications for shale gas production. The system involves highly complex geo-textures …

Splitting method is a powerful method to handle application problems by splitting physics,
scales, domain, and so on. Many splitting algorithms have been designed for efficient …