Related papers: Weak TransNet: A Petrov-Galerkin based neural netw…

This paper presents the Tensor Product Network (TPNet), a novel neural architecture for efficient and accurate function approximation and PDE solving. The core of the proposal involves constructing the solution explicitly as a linear…

We propose ARDO method for solving PDEs and PDE-related problems with deep learning techniques. This method uses a weak adversarial formulation but transfers the random difference operator onto the test function. The main advantage of this…

Transfer learning for partial differential equations (PDEs) is to develop a pre-trained neural network that can be used to solve a wide class of PDEs. Existing transfer learning approaches require much information of the target PDEs such as…

This paper proposes a weak Galerkin (WG) finite element method for elliptic interface problems defined on nonconvex polygonal partitions. The method features a built-in stabilizer and retains a simple, symmetric, and positive definite…

The transferable neural network (TransNet) is a two-layer shallow neural network with pre-determined and uniformly distributed neurons in the hidden layer, and the least-squares solvers can be particularly used to compute the parameters of…

We present a subspace method based on neural networks for solving the partial differential equation in weak form with high accuracy. The basic idea of our method is to use some functions based on neural networks as base functions to span a…

Physics-informed neural networks (PINNs) [31] use automatic differentiation to solve partial differential equations (PDEs) by penalizing the PDE in the loss function at a random set of points in the domain of interest. Here, we develop a…

This paper presents a new weak Galerkin (WG) method for elliptic interface problems on general curved polygonal partitions. The method's key innovation lies in its ability to transform the complex interface jump condition into a more…

Deep learning has emerged as a compelling framework for scientific and engineering computing, motivating growing interest in neural network-based solvers for partial differential equations (PDEs). Within this landscape, network…

Machine learning methods have been lately used to solve partial differential equations (PDEs) and dynamical systems. These approaches have been developed into a novel research field known as scientific machine learning in which techniques…

Linearized shallow neural networks that are constructed by fixing the hidden-layer parameters have recently shown strong performance in solving partial differential equations (PDEs). Such models, widely used in the random feature method…

Petrov-Galerkin formulations with optimal test functions allow for the stabilization of finite element simulations. In particular, given a discrete trial space, the optimal test space induces a numerical scheme delivering the best…

The novel idea of weak Galerkin (WG) finite element methods is on the use of weak functions and their weak derivatives defined as distributions. Weak functions and weak derivatives can be approximated by polynomials with various degrees.…

The weak Galerkin (WG) finite element method is an effective and flexible general numerical technique for solving partial differential equations. It is a natural extension of the classic conforming finite element method for discontinuous…

A weak Galerkin (WG) finite element method for solving the stationary Stokes equations in two- or three- dimensional spaces by using discontinuous piecewise polynomials is developed and analyzed. The variational form we considered is based…

The weak Galerkin finite element method is a novel numerical method that was first proposed and analyzed by Wang and Ye for general second order elliptic problems on triangular meshes. The goal of this paper is to conduct a computational…

We develop a new $C^{\,0}$-continuous Petrov-Galerkin spectral element method for one-dimensional fractional elliptic problems of the form ${}_{0}{\mathcal{D}}_{x}^{\alpha} u(x) - \lambda u(x) = f(x)$, $\alpha \in (1,2]$, subject to…

The weak Galerkin (WG) finite element method is an effective and flexible general numerical technique for solving partial differential equations. The novel idea of weak Galerkin finite element methods is on the use of weak functions and…

To overcome these obstacles and improve computational accuracy and efficiency, this paper presents the Randomized Radial Basis Function Neural Network (RRNN), an innovative approach explicitly crafted for solving multiscale elliptic…

In this article, a weak Galerkin method is firstly presented and analyzed for the quasi-linear elliptic problem of non-monotone type. By using Brouwer's fixed point technique, the existence of WG solution and error estimates in both the…