Related papers: Multi-scale Deep Neural Network (MscaleDNN) for So…
On the forefront of scientific computing, Deep Learning (DL), i.e., machine learning with Deep Neural Networks (DNNs), has emerged a powerful new tool for solving Partial Differential Equations (PDEs). It has been observed that DNNs are…
A tandem deep neural network approach is presented for the inverse design of reactively loaded metasurfaces with prescribed far-field radiation characteristics. The proposed approach integrates a deep neural network (DNN) with a…
We consider learning deep neural networks (DNNs) that consist of low-precision weights and activations for efficient inference of fixed-point operations. In training low-precision networks, gradient descent in the backward pass is performed…
In recent work it has been established that deep neural networks are capable of approximating solutions to a large class of parabolic partial differential equations without incurring the curse of dimension. However, all this work has been…
Partial Differential Equations (PDEs) are integral to modeling many scientific and engineering problems. Physics-informed Neural Networks (PINNs) have emerged as promising tools for solving PDEs by embedding governing equations into the…
Regression with non-Euclidean responses -- e.g., probability distributions, networks, symmetric positive-definite matrices, and compositions -- has become increasingly important in modern applications. In this paper, we propose deep…
A unified deep neural network, denoted the multi-scale CNN (MS-CNN), is proposed for fast multi-scale object detection. The MS-CNN consists of a proposal sub-network and a detection sub-network. In the proposal sub-network, detection is…
The aim of this work is to consider multiscale algorithms for solving PDEs with Galerkin methods on bounded domains. We provide results on convergence and condition numbers. We show how to handle PDEs with Dirichlet boundary conditions. We…
In this article, we investigate the existence of a deep neural network (DNN) capable of approximating solutions to partial integro-differential equations while circumventing the curse of dimensionality. Using the Feynman-Kac theorem, we…
This paper introduces Progressively Diffused Networks (PDNs) for unifying multi-scale context modeling with deep feature learning, by taking semantic image segmentation as an exemplar application. Prior neural networks, such as ResNet, tend…
We propose a new approach, called as functional deep neural network (FDNN), for classifying multi-dimensional functional data. Specifically, a deep neural network is trained based on the principle components of the training data which shall…
Spectral methods are an important part of scientific computing's arsenal for solving partial differential equations (PDEs). However, their applicability and effectiveness depend crucially on the choice of basis functions used to expand the…
The numerical solution of high dimensional partial differential equations (PDEs) is severely constrained by the curse of dimensionality (CoD), rendering classical grid--based methods impractical beyond a few dimensions. In recent years,…
Recently, deep neural networks (DNNs) have become powerful tools for solving inverse scattering problems. However, the approximation and generalization rates of DNNs for solving these problems remain largely under-explored. In this work, we…
The intrinsically infinite-dimensional features of the functional observations over multidimensional domains render the standard classification methods effectively inapplicable. To address this problem, we introduce a novel multiclass…
Neural networks are powerful tools for approximating high dimensional data that have been used in many contexts, including solution of partial differential equations (PDEs). We describe a solver for multiscale fully nonlinear elliptic…
Semilinear parabolic partial differential equations (PDEs) are fundamental to modeling complex dynamical systems across scientific domains. The Deep Backward Stochastic Differential Equation (BSDE) method is a promising approach for…
We introduce a deep residual recurrent neural network (DR-RNN) as an efficient model reduction technique for nonlinear dynamical systems. The developed DR-RNN is inspired by the iterative steps of line search methods in finding the residual…
Deep neural networks (DNNs) are increasingly used to solve partial differential equations (PDEs) that naturally arise while modeling a wide range of systems and physical phenomena. However, the accuracy of such DNNs decreases as the PDE…
We propose a new class of Bayesian neural networks (BNNs) that can be trained using noisy data of variable fidelity, and we apply them to learn function approximations as well as to solve inverse problems based on partial differential…