Related papers: Deep Euler method: solving ODEs by approximating t…

Improved local numerical solution for the ADER-DG numerical method with a local DG predictor for solving the initial value problem for a first-order ODE system is proposed. The improved local numerical solution demonstrates convergence…

Machine learning methods have been successful in many areas, like image classification and natural language processing. However, it still needs to be determined how to apply ML to areas with mathematical constraints, like solving PDEs.…

Artificial intelligence and deep learning are currently reshaping numerical simulation frameworks by introducing new modeling capabilities. These frameworks are extensively investigated in the context of model correction and…

Ordinary and partial differential equations (DE) are used extensively in scientific and mathematical domains to model physical systems. Current literature has focused primarily on deep neural network (DNN) based methods for solving a…

Solving high-dimensional partial differential equations (PDEs) is a major challenge in scientific computing. We develop a new numerical method for solving elliptic-type PDEs by adapting the Q-learning algorithm in reinforcement learning.…

There has been significant recent interest in the use of deep learning for regularizing imaging inverse problems. Most work in the area has focused on regularization imposed implicitly by convolutional neural networks (CNNs) pre-trained for…

The primary goal of this research is to propose a novel architecture for a deep neural network that can solve fractional differential equations accurately. A Gaussian integration rule and a $L_1$ discretization technique are used in the…

In this paper, based on the combination of tensor neural network and a posteriori error estimator, a novel type of machine learning method is proposed to solve high-dimensional boundary value problems with homogeneous and non-homogeneous…

A state-of-the-art deep domain decomposition method (D3M) based on the variational principle is proposed for partial differential equations (PDEs). The solution of PDEs can be formulated as the solution of a constrained optimization…

In this paper, we demonstrate a computationally efficient new approach based on deep learning (DL) techniques for analysis, design, and optimization of electromagnetic (EM) nanostructures. We use the strong correlation among features of a…

On the one hand, the explicit Euler scheme fails to converge strongly to the exact solution of a stochastic differential equation (SDE) with a superlinearly growing and globally one-sided Lipschitz continuous drift coefficient. On the other…

This paper presents a proper generalized decomposition (PGD) based reduced-order model of hierarchical deep-learning neural networks (HiDeNN). The proposed HiDeNN-PGD method keeps both advantages of HiDeNN and PGD methods. The automatic…

We analyse errors of randomized explicit and implicit Euler schemes for approximate solving of ordinary differential equations (ODEs). We consider classes of ODEs for which the right-hand side functions satisfy Lipschitz condition globally…

Recently, Li et al. (Bioinformatics 27(19), 2686-91, 2011) proposed a method, called Differential Equation-based Local Dynamic Bayesian Network (DELDBN), for reverse engineering gene regulatory networks from time-course data. We commend the…

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…

ODENet is a deep neural network architecture in which a stacking structure of ResNet is implemented with an ordinary differential equation (ODE) solver. It can reduce the number of parameters and strike a balance between accuracy and…

Many important problems in science and engineering require solving the so-called parametric partial differential equations (PDEs), i.e., PDEs with different physical parameters, boundary conditions, shapes of computation domains, etc.…

In this paper we deal with global approximation of solutions of stochastic differential equations (SDEs) driven by countably dimensional Wiener process. Under certain regularity conditions imposed on the coefficients, we show lower bounds…

We propose an algorithm for approximating the solution of a strongly oscillating SDE, that is, a system in which some ergodic state variables evolve quickly with respect to the other variables. The algorithm profits from homogenization…

This paper focuses on explicit approximations for nonlinear stochastic delay differential equations (SDDEs). Under the weakly local Lipschitz and some suitable conditions, a generic truncated Euler-Maruyama (TEM) scheme for SDDEs is…