相关论文: Neural Network approximation power on homogeneous …

This paper focuses on establishing $L^2$ approximation properties for deep ReLU convolutional neural networks (CNNs) in two-dimensional space. The analysis is based on a decomposition theorem for convolutional kernels with a large spatial…

We present approximation results and numerical experiments for the use of randomized neural networks within physics-informed extreme learning machines to efficiently solve high-dimensional PDEs, demonstrating both high accuracy and low…

Nonlinear differential equations are challenging to solve numerically and are important to understanding the dynamics of many physical systems. Deep neural networks have been applied to help alleviate the computational cost that is…

While deep learning is successful in a number of applications, it is not yet well understood theoretically. A satisfactory theoretical characterization of deep learning however, is beginning to emerge. It covers the following questions: 1)…

Diffusion processes have been widely used for approximations in the queueing theory. There are different types of diffusion approximations. Among them, we are interested in those obtained through limits of a sequence of models which…

Deep neural networks are increasingly being used in cognitive modeling as a means of deriving representations for complex stimuli such as images. While the predictive power of these networks is high, it is often not clear whether they also…

This paper is concerned with the utilization of deterministically modeled chemical reaction networks for the implementation of (feed-forward) neural networks. We develop a general mathematical framework and prove that the ordinary…

In this paper we present a method to approximate optimal feedback controls for stochastic reaction-diffusion equations. We derive two approximation results providing the theoretical foundation of our approach and allowing for explicit error…

This paper considers one of the fundamental parallel-in-time methods for the solution of ordinary differential equations, Parareal, and extends it by adopting a neural network as a coarse propagator. We provide a theoretical analysis of the…

In this paper, neural network approximation methods are developed for elliptic partial differential equations with multi-frequency solutions. Neural network work approximation methods have advantages over classical approaches in that they…

In this paper, we study the partial differential equation models of neural networks. Neural network can be viewed as a map from a simple base model to a complicate function. Based on solid analysis, we show that this map can be formulated…

Neural networks, a central tool in machine learning, have demonstrated remarkable, high fidelity performance on image recognition and classification tasks. These successes evince an ability to accurately represent high dimensional…

This paper is an introduction to the membrane potential equation for neurons. Its properties are described, as well as sample applications. Networks of these equations can be used for modeling neuronal systems, which also process images and…

We describe generalizations of the universal approximation theorem for neural networks to maps invariant or equivariant with respect to linear representations of groups. Our goal is to establish network-like computational models that are…

Feedforward neural networks have wide applicability in various disciplines of science due to their universal approximation property. Some authors have shown that single hidden layer feedforward neural networks (SLFNs) with fixed weights…

Recently, deep learning has achieved huge successes in many important applications. In our previous studies, we proposed quadratic/second-order neurons and deep quadratic neural networks. In a quadratic neuron, the inner product of a vector…

Deep learning has been widely applied and brought breakthroughs in speech recognition, computer vision, and many other domains. The involved deep neural network architectures and computational issues have been well studied in machine…

In the mean field regime, neural networks are appropriately scaled so that as the width tends to infinity, the learning dynamics tends to a nonlinear and nontrivial dynamical limit, known as the mean field limit. This lends a way to study…

We present a simple, yet general, end-to-end deep neural network representation of the potential energy surface for atomic and molecular systems. This methodology, which we call Deep Potential, is "first-principle" based, in the sense that…

Molecular modeling is an important topic in drug discovery. Decades of research have led to the development of high quality scalable molecular force fields. In this paper, we show that neural networks can be used to train a universal…