Related papers: A Shallow Ritz Method for Elliptic Problems with S…

Several important problems in partial differential equations can be formulated as integral equations. Often the integral operator defines the solution of an elliptic problem with specified jump conditions at an interface. In principle the…

In this paper, we propose a novel shape optimization approach for the source identification of elliptic equations. This identification problem arises from two application backgrounds: actuator placement in PDE-constrained optimal controls…

Neural networks are powerful tools for approximating high dimensional data that have been used in many contexts, including solution of partial differential equations (PDEs). We describe a solver for multiscale fully nonlinear elliptic…

Residual minimization is a widely used technique for solving Partial Differential Equations in variational form. It minimizes the dual norm of the residual, which naturally yields a saddle-point (min-max) problem over the so-called trial…

We propose a mesh-free method to solve nonconvex energy minimization problems for martensitic phase transitions and twinning in crystals, using the deep learning approach. These problems pose multiple challenges to both analysis and…

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)…

We consider a slightly subcritical Dirichlet problem with a non-power nonlinearity in a bounded smooth domain. For this problem, standard compact embeddings cannot be used to guarantee the existence of solutions as in the case of power-type…

Neural network approaches have been demonstrated to work quite well to solve partial differential equations in practice. In this context approaches like physics-informed neural networks and the Deep Ritz method have become popular. In this…

In this paper, we study the statistical limits of deep learning techniques for solving elliptic partial differential equations (PDEs) from random samples using the Deep Ritz Method (DRM) and Physics-Informed Neural Networks (PINNs). To…

This paper proposes a new parametric level set method for topology optimization based on Deep Neural Network (DNN). In this method, the fully connected deep neural network is incorporated into the conventional level set methods to construct…

The singularities that arise in elliptic boundary value problems are treated locally by a singular function boundary integral method. This method extracts the leading singular coefficients from a series expansion that describes the local…

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…

The neural network method of solving differential equations is used to approximate the electric potential and corresponding electric field in the slit-well microfluidic device. The device's geometry is non-convex, making this a challenging…

We develop a weak adversarial approach to solving obstacle problems using neural networks. By employing (generalised) regularised gap functions and their properties we rewrite the obstacle problem (which is an elliptic variational…

This paper proposes a deep-learning-based domain decomposition method (DeepDDM), which leverages deep neural networks (DNN) to discretize the subproblems divided by domain decomposition methods (DDM) for solving partial differential…

In this work, we investigate a neural network based solver for optimal control problems (without / with box constraint) for linear and semilinear second-order elliptic problems. It utilizes a coupled system derived from the first-order…

We introduce an $r-$adaptive algorithm to solve Partial Differential Equations using a Deep Neural Network. The proposed method restricts to tensor product meshes and optimizes the boundary node locations in one dimension, from which we…

Recent years have witnessed growing interests in solving partial differential equations by deep neural networks, especially in the high-dimensional case. Unlike classical numerical methods, such as finite difference method and finite…

With the remarkable empirical success of neural networks across diverse scientific disciplines, rigorous error and convergence analysis are also being developed and enriched. However, there has been little theoretical work focusing on…

We propose a deep learning based method, the Deep Ritz Method, for numerically solving variational problems, particularly the ones that arise from partial differential equations. The Deep Ritz method is naturally nonlinear, naturally…