Related papers: Deep Ritz revisited

We propose rigorous lower and upper error bounds for neural network (NN) approximations to PDEs by efficiently computing the Riesz representations of suitable extension and restrictions of the NN residual towards geometrically simpler…

Physics-informed Neural Networks (PINNs) have been shown to be effective in solving partial differential equations by capturing the physics induced constraints as a part of the training loss function. This paper shows that a PINN can be…

In this work, we study physics-informed neural networks (PINNs) constrained by partial differential equations (PDEs) and their application in approximating PDEs with two characteristic scales. From a continuous perspective, our formulation…

We introduce the Optimizing a Discrete Loss (ODIL) framework for the numerical solution of Partial Differential Equations (PDE) using machine learning tools. The framework formulates numerical methods as a minimization of discrete residuals…

In this work we present an application of modern deep learning methodologies to the numerical solution of partial differential equations in transport models. More specifically, we employ a supervised deep neural network that takes into…

We consider the problem of numerically approximating the solutions to a partial differential equation (PDE) when there is insufficient information to determine a unique solution. Our main example is the Poisson boundary value problem, when…

We present a novel deep learning approach to approximate the solution of large, sparse, symmetric, positive-definite linear systems of equations. These systems arise from many problems in applied science, e.g., in numerical methods for…

Deep neural networks (DNNs), especially physics-informed neural networks (PINNs), have recently become a new popular method for solving forward and inverse problems governed by partial differential equations (PDEs). However, these methods…

In this work, we develop an efficient solver based on neural networks for second-order elliptic equations with variable coefficients and singular sources. This class of problems covers general point sources, line sources and the combination…

This paper focuses on the problem of Differentially Private Stochastic Optimization for (multi-layer) fully connected neural networks with a single output node. In the first part, we examine cases with no hidden nodes, specifically focusing…

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…

I provide an introduction to the application of deep learning and neural networks for solving partial differential equations (PDEs). The approach, known as physics-informed neural networks (PINNs), involves minimizing the residual of the…

In this paper we study probabilistic and neural network approximations for solutions to Poisson equation subject to Holder data in general bounded domains of $\mathbb{R}^d$. We aim at two fundamental goals. The first, and the most…

Recently established equivalences between differential equations and the structure of neural networks enabled some interpretation of training of a neural network as partial-differential-equation (PDE) constrained optimization. We add to the…

The reliable and accurate numerical approximation of the $p$-Laplacian is particularly challenging in the extreme regimes $p \to 1^{+}$ and $p \gg 1$, where the operator becomes either highly singular or strongly degenerate, often causing…

The present study investigates a linear-quadratic Dirichlet control problem governed by a non-coercive elliptic equation posed on a possibly non-convex polygonal domain. Tikhonov regularization is carried out in an energy seminorm. The…

We develop a numerical strategy to solve multi-dimensional Poisson equations on dynamically adapted grids for evolutionary problems disclosing propagating fronts. The method is an extension of the multiresolution finite volume scheme used…

Learning the solution of partial differential equations (PDEs) with a neural network is an attractive alternative to traditional solvers due to its elegance, greater flexibility and the ease of incorporating observed data. However, training…

We consider the fractional elliptic problem with Dirichlet boundary conditions on a bounded and convex domain $D$ of $\mathbb{R}^d$, with $d \geq 2$. In this paper, we perform a stochastic gradient descent algorithm that approximates the…

Partial differential equations (PDEs) are indispensable for modeling many physical phenomena and also commonly used for solving image processing tasks. In the latter area, PDE-based approaches interpret image data as discretizations of…