相关论文: DGNN: A Neural PDE Solver Induced by Discontinuous…

This paper develops and analyzes an interior penalty discontinuous Galerkin (IPDG) method using piecewise linear polynomials for the indefinite time harmonic Maxwell equations with the impedance boundary condition in the three dimensional…

For hyperbolic conservation laws, traditional methods and physics-informed neural networks (PINNs) often encounter difficulties in capturing sharp discontinuities and maintaining temporal consistency. To address these challenges, we…

Over the past few years, neural network methods have evolved in various directions for approximating partial differential equations (PDEs). A promising new development is the integration of neural networks with classical numerical…

Discontinuous Galerkin (DG) methods for the numerical solution of partial differential equations have enjoyed considerable success because they are both flexible and robust: They allow arbitrary unstructured geometries and easy control of…

The discontinuous Galerkin (DG) algorithm is a representative high order method in Computational Fluid Dynamics (CFD) area which possesses considerable mathematical advantages such as high resolution, low dissipation, and dispersion.…

This paper constitutes our initial effort in developing sparse grid discontinuous Galerkin (DG) methods for high-dimensional partial differential equations (PDEs). Over the past few decades, DG methods have gained popularity in many…

In this work we apply the Deep Galerkin Method (DGM) described in Sirignano and Spiliopoulos (2018) to solve a number of partial differential equations that arise in quantitative finance applications including option pricing, optimal…

The diffusive-viscous wave equation is an advancement in wave equation theory, as it accounts for both diffusion and viscosity effects. This has a wide range of applications in geophysics, such as the attenuation of seismic waves in…

We propose and analyze discontinuous Galerkin (dG) approximations to 3D-1D coupled systems which model diffusion in a 3D domain containing a small inclusion reduced to its 1D centerline. Convergence to weak solutions of a steady state…

In this paper, we consider a class of discontinuous Galerkin (DG) methods for one-dimensional nonlocal diffusion (ND) problems. The nonlocal models, which are integral equations, are widely used in describing many physical phenomena with…

This work considers stochastic Galerkin approximations of linear elliptic partial differential equations (PDEs) with stochastic forcing terms and stochastic diffusion coefficients, that cannot be bounded uniformly away from zero and…

Physics informed neural networks (PINNs) have emerged as a powerful tool to provide robust and accurate approximations of solutions to partial differential equations (PDEs). However, PINNs face serious difficulties and challenges when…

We extend the discontinuous Galerkin (DG) framework to a linear second-order elliptic problem on a compact smooth connected and oriented surface. An interior penalty (IP) method is introduced on a discrete surface and we derive a-priori…

This paper develops interior penalty discontinuous Galerkin (IP-DG) methods to approximate $W^{2,p}$ strong solutions of second order linear elliptic partial differential equations (PDEs) in non-divergence form with continuous coefficients.…

There has been an arising trend of adopting deep learning methods to study partial differential equations (PDEs). This article is to propose a Deep Learning Galerkin Method (DGM) for the closed-loop geothermal system, which is a new coupled…

We introduce an $hp$-version symmetric interior penalty discontinuous Galerkin finite element method (DGFEM) for the numerical approximation of the biharmonic equation on general computational meshes consisting of polygonal/polyhedral…

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…

We extend the Deep Galerkin Method (DGM) introduced in Sirignano and Spiliopoulos (2018)} to solve a number of partial differential equations (PDEs) that arise in the context of optimal stochastic control and mean field games. First, we…

In this paper, we study deep neural networks (DNNs) for solving high-dimensional evolution equations with oscillatory solutions. Different from deep least-squares methods that deal with time and space variables simultaneously, we propose a…

The interior penalty discontinuous Galerkin method is applied to solve elliptic equations on either networks of segments or networks of planar surfaces, with arbitrary but fixed number of bifurcations. Stability is obtained by proving a…