Related papers: Prediction of numerical homogenization using deep …

We develop a minimax rate analysis to describe the reason that deep neural networks (DNNs) perform better than other standard methods. For nonparametric regression problems, it is well known that many standard methods attain the minimax…

We present an adaptive regularization algorithm that can be effectively applied to the optimization problem in deep learning framework. Our regularization algorithm aims to take into account the fitness of data to the current state of model…

Multiscale problems can usually be approximated through numerical homogenization by an equation with some effective parameters that can capture the macroscopic behavior of the original system on the coarse grid to speed up the simulation.…

The rapid development of deep learning has significant implications for the advancement of Computational Fluid Dynamics (CFD). Currently, most pixel-grid-based deep learning methods for flow field prediction exhibit significantly reduced…

Using deep neural networks to solve PDEs has attracted a lot of attentions recently. However, why the deep learning method works is falling far behind its empirical success. In this paper, we provide a rigorous numerical analysis on deep…

In this paper, we propose a deep learning based reduced order modeling method for stochastic underground flow problems in highly heterogeneous media. We aim to utilize supervised learning to build a reduced surrogate model from the…

This paper develops a randomized approach for incrementally building deep neural networks, where a supervisory mechanism is proposed to constrain the random assignment of the weights and biases, and all the hidden layers have direct links…

In this study, we focus on the modelling of infiltration process in porous media. We use the meshless techniques for efficiently solving the Richards equation which describes unsaturated water flow through soils. The design of approximate…

Uncertainty propagation in nonlinear dynamic systems remains an outstanding problem in scientific computing and control. Numerous approaches have been developed, but are limited in their capability to tackle problems with more than a few…

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

In comparison to classical shallow representation learning techniques, deep neural networks have achieved superior performance in nearly every application benchmark. But despite their clear empirical advantages, it is still not well…

Deep learning has been wildly successful in practice and most state-of-the-art machine learning methods are based on neural networks. Lacking, however, is a rigorous mathematical theory that adequately explains the amazing performance of…

Despite the high performance achieved by deep neural networks on various tasks, extensive studies have demonstrated that small tweaks in the input could fail the model predictions. This issue of deep neural networks has led to a number of…

Standard deep neural networks (DNNs) are commonly trained in an end-to-end fashion for specific tasks such as object recognition, face identification, or character recognition, among many examples. This specificity often leads to…

Heterogeneous Graph Neural Networks (HGNNs) are a class of deep learning models designed specifically for heterogeneous graphs, which are graphs that contain different types of nodes and edges. This paper investigates the application of…

We describe the numerical scheme for the discretization and solution of 2D elliptic equations with strongly varying piecewise constant coefficients arising in the stochastic homogenization of multiscale composite materials. An efficient…

Traditional approaches based on finite element analyses have been successfully used to predict the macro-scale behavior of heterogeneous materials (composites, multicomponent alloys, and polycrystals) widely used in industrial applications.…

Deep networks realize complex mappings that are often understood by their locally linear behavior at or around points of interest. For example, we use the derivative of the mapping with respect to its inputs for sensitivity analysis, or to…

The fast and accurate prediction of unsteady flow becomes a serious challenge in fluid dynamics, due to the high-dimensional and nonlinear characteristics. A novel hybrid deep neural network (DNN) architecture was designed to capture the…

Supervised training of deep neural nets typically relies on minimizing cross-entropy. However, in many domains, we are interested in performing well on metrics specific to the application. In this paper we propose a direct loss minimization…