Related papers: A multi-scale DNN algorithm for nonlinear elliptic…

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…

Deep neural networks (DNN) has received increasing attention in machine learning applications in the last several years. Recently, a non-asymptotic error bound has been developed to measure the performance of the fully connected DNN…

Deep convolutional neural networks (DCNN) have enjoyed great successes in many signal processing applications because they can learn complex, non-linear causal relationships from input to output. In this light, DCNNs are well suited for the…

Nonlinear regression problem is one of the most popular and important statistical tasks. The first methods like least squares estimation go back to Gauss and Legendre. Recent models and developments in statistics and machine learning like…

Complex nonlinear models such as deep neural network (DNNs) have become an important tool for image classification, speech recognition, natural language processing, and many other fields of application. These models however lack…

Motivated by recent research on Physics-Informed Neural Networks (PINNs), we make the first attempt to introduce the PINNs for numerical simulation of the elliptic Partial Differential Equations (PDEs) on 3D manifolds. PINNs are one of the…

Artificial intelligence is revolutionizing our lives at an ever increasing pace. At the heart of this revolution is the recent advancements in deep neural networks (DNN), learning to perform sophisticated, high-level tasks. However,…

In this paper, the authors propose the utilization of Fibonacci Neural Networks (FNN) for solving arbitrary order differential equations. The FNN architecture comprises input, middle, and output layers, with various degrees of Fibonacci…

Deep neural networks (DNNs) have been used to model complex optimization problems in many applications, yet have difficulty guaranteeing solution optimality and feasibility, despite training on large datasets. Training a NN as a surrogate…

Discrete time stochastic optimal control problems and Markov decision processes (MDPs) are fundamental models for sequential decision-making under uncertainty and as such provide the mathematical framework underlying reinforcement learning…

We propose in this work the gradient-enhanced deep neural networks (DNNs) approach for function approximations and uncertainty quantification. More precisely, the proposed approach adopts both the function evaluations and the associated…

Physics-informed holomorphic neural networks (PIHNNs) have recently emerged as efficient surrogate models for solving differential problems. By embedding the underlying problem structure into the network, PIHNNs require training only to…

A main theoretical interest in biology and physics is to identify the nonlinear dynamical system (DS) that generated observed time series. Recurrent Neural Networks (RNNs) are, in principle, powerful enough to approximate any underlying DS,…

In this paper, a meshfree method using the deep neural network (DNN) approach is developed for solving two kinds of dynamic two-phase interface problems governed by different dynamic partial differential equations on either side of the…

Deep Neural Networks (DNNs) have achieved state-of-the-art accuracy performance in many tasks. However, recent works have pointed out that the outputs provided by these models are not well-calibrated, seriously limiting their use in…

We investigate scaling and efficiency of the deep neural network multigrid method (DNN-MG). DNN-MG is a novel neural network-based technique for the simulation of the Navier-Stokes equations that combines an adaptive geometric multigrid…

Deep neural networks have garnered widespread attention due to their simplicity and flexibility in the fields of engineering and scientific calculation. In this study, we probe into solving a class of elliptic partial differential…

This paper introduces a novel algorithmic framework for a deep neural network (DNN), which in a mathematically rigorous manner, allows us to incorporate history (or memory) into the network -- it ensures all layers are connected to one…

The problem of automatic and accurate forecasting of time-series data has always been an interesting challenge for the machine learning and forecasting community. A majority of the real-world time-series problems have non-stationary…

Learning time-dependent partial differential equations (PDEs) that govern evolutionary observations is one of the core challenges for data-driven inference in many fields. In this work, we propose to capture the essential dynamics of…