相关论文: A Priori Generalization Analysis of the Deep Ritz …

In recent years, there has been a growing interest in leveraging deep learning and neural networks to address scientific problems, particularly in solving partial differential equations (PDEs). However, many neural network-based methods…

We establish error estimates for the approximation of parametric $p$-Dirichlet problems deploying the Deep Ritz Method. Parametric dependencies include, e.g., varying geometries and exponents $p\in (1,\infty)$. Combining the derived error…

We investigate the multi-dimensional Super Resolution problem on closed semi-algebraic domains for various sampling schemes such as Fourier or moments. We present a new semidefinite programming (SDP) formulation of the 1 -minimization in…

In this paper, we propose a semigroup method for solving high-dimensional elliptic partial differential equations (PDEs) and the associated eigenvalue problems based on neural networks. For the PDE problems, we reformulate the original…

The purpose of this work is the study of solution techniques for problems involving fractional powers of symmetric coercive elliptic operators in a bounded domain with Dirichlet boundary conditions. These operators can be realized as the…

Numerically solving high-dimensional partial differential equations (PDEs) is a major challenge. Conventional methods, such as finite difference methods, are unable to solve high-dimensional PDEs due to the curse-of-dimensionality. A…

We use elliptic partial differential equations (PDEs) as examples to show various properties and behaviors when shallow neural networks (SNNs) are used to represent the solutions. In particular, we study the numerical ill-conditioning,…

A universal rule-based self-learning approach using deep reinforcement learning (DRL) is proposed for the first time to solve nonlinear ordinary differential equations and partial differential equations. The solver consists of a deep neural…

The direct deep learning simulation for multi-scale problems remains a challenging issue. In this work, a novel higher-order multi-scale deep Ritz method (HOMS-DRM) is developed for thermal transfer equation of authentic composite materials…

In this paper, we introduce the Deep Finite Volume Method (DFVM), an innovative deep learning framework tailored for solving high-order (order \(\geq 2\)) partial differential equations (PDEs). Our approach centers on a novel loss function…

In this paper, we study the deep Ritz method for solving the linear elasticity equation from a numerical analysis perspective. A modified Ritz formulation using the $H^{1/2}(\Gamma_D)$ norm is introduced and analyzed for linear elasticity…

We compare different training strategies for the Deep Ritz Method for elliptic equations with Dirichlet boundary conditions and highlight the problems arising from the boundary values. We distinguish between an exact resolution of the…

Machine learning has been successfully applied to various fields of scientific computing in recent years. In this work, we propose a sparse radial basis function neural network method to solve elliptic partial differential equations (PDEs)…

This article proposes a hybrid adaptive numerical method based on the Dual Reciprocity Method (DRM) to solve problems with non-linear boundary conditions and large-scale problems, named Hybrid Adaptive Dual Reciprocity Method (H-DRM). The…

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…

Partial differential equations (PDEs) are widely used for modeling various physical phenomena. These equations often depend on certain parameters, necessitating either the identification of optimal parameters or the solution of the…

We study the generalized forward-reflected-backward (GFRB) method, an extension of the forward-reflected-backward (FRB) scheme due to Malitsky and Tam, for solving monotone inclusion problems in real Hilbert spaces. We first analyze GFRB…

The discretization of elliptic PDEs leads to large coupled systems of equations. Domain decomposition methods (DDMs) are one approach to the solution of these systems, and can split the problem in a way that allows for parallel computing.…

One of the oldest and most studied subject in scientific computing is algorithms for solving partial differential equations (PDEs). A long list of numerical methods have been proposed and successfully used for various applications. In…

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…