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Neural networks have shown significant potential in solving partial differential equations (PDEs). While deep networks are capable of approximating complex functions, direct one-shot training often faces limitations in both accuracy and…

We analyze a hybrid method that enriches coarse grid finite element solutions with fine scale fluctuations obtained from a neural network. The idea stems from the Deep Neural Network Multigrid Solver (DNN-MG), (Margenberg et al., J Comput…

In practice, deep neural networks are often able to easily interpolate their training data. To understand this phenomenon, many works have aimed to quantify the memorization capacity of a neural network architecture: the largest number of…

Level set method has been used to capture interface motion. Narrow band algorithm is applied to localize the solving of level-set PDE on global domain to a tube around interface. Due to the unknown evolving interface, narrow band algorithm…

Block-to-block interface interpolation operators are constructed for several common high-order finite difference discretizations. In contrast to conventional interpolation operators, these new interpolation operators maintain the strict…

In this paper we explore acceleration techniques for large scale nonconvex optimization problems with special focuses on deep neural networks. The extrapolation scheme is a classical approach for accelerating stochastic gradient descent for…

In the context of classification problems, Deep Learning (DL) approaches represent state of art. Many DL approaches are based on variations of standard multi-layer feed-forward neural networks. These are also referred to as deep networks.…

In this manuscript, we present a collective multigrid algorithm to solve efficiently the large saddle-point systems of equations that typically arise in PDE-constrained optimization under uncertainty, and develop a novel convergence…

This paper provides a theoretical framework on the solution of feedforward ReLU networks for interpolations, in terms of what is called an interpolation matrix, which is the summary, extension and generalization of our three preceding…

In the context of deep learning, this article presents an original deep network, namely CentralNet, for the fusion of information coming from different sensors. This approach is designed to efficiently and automatically balance the…

We propose a new deflation strategy to accelerate the convergence of the preconditioned conjugate gradient(PCG) method for solving parametric large-scale linear systems of equations. Unlike traditional deflation techniques that rely on…

The use of multigrid and related preconditioners with the finite element method is often limited by the difficulty of applying the algorithm effectively to a problem, especially when the domain has a complex shape or adaptive refinement. We…

Matrix-free geometric multigrid solvers for elliptic PDEs that have been discretised with Higher-order Discontinuous Galerkin (DG) methods are ideally suited to exploit state-of-the-art computer architectures. Higher polynomial degrees…

We propose a simple and efficient one-way multigrid method for self-consistent electronic structure calculations based on iterative diagonalization. Total energy calculations are performed on several different levels of grids starting from…

While many problems in machine learning focus on learning mappings between finite-dimensional spaces, scientific applications require approximating mappings between function spaces, i.e., operators. We study the problem of learning…

We present a tailored multigrid method for linear problems stemming from a Nitsche-based extended finite element method (XFEM). Our multigrid method is robust with respect to highly varying coefficients and the number of interfaces in a…

We propose a multilevel approach for trace systems resulting from hybridized discontinuous Galerkin (HDG) methods. The key is to blend ideas from nested dissection, domain decomposition, and high-order characteristic of HDG discretizations.…

This paper introduces a novel approach to algebraic multigrid methods for large systems of linear equations coming from finite element discretizations of certain elliptic second order partial differential equations. Based on a discrete…

Scientific computing using deep learning has seen significant advancements in recent years. There has been growing interest in models that learn the operator from the parameters of a partial differential equation (PDE) to the corresponding…

Complex Networks are a good approach to find internal relationships and represent the structure of classes in a dataset then they are used for High Level Classification. Previous works use K-Nearest Neighbors to build each Complex Network…