Related papers: Multi-patch isogeometric neural solver for partial…
Physics-Informed Neural Networks (PINNs) are a powerful class of numerical solvers for partial differential equations, employing deep neural networks with successful applications across a diverse set of problems. However, their…
In recent years, machine learning and deep learning approaches such as artificial neural networks have gained in popularity for the resolution of automatic puzzle resolution problems. Indeed, these methods are able to extract high-level…
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…
This paper aims to devise an adaptive neural network basis method for numerically solving a second-order semilinear partial differential equation (PDE) with low-regular solutions in two/three dimensions. The method is obtained by combining…
Recently deep learning surrogates and neural operators have shown promise in solving partial differential equations (PDEs). However, they often require a large amount of training data and are limited to bounded domains. In this work, we…
One of the reasons for the success of the finite element method is its versatility to deal with different types of geometries. This is particularly true of problems posed in curved domains of arbitrary shape. In the case of second order…
Simulation and modeling are essential in product development, integrated into the design and manufacturing process to enhance efficiency and quality. They are typically represented as complex nonlinear differential algebraic equations. The…
A state-of-the-art deep domain decomposition method (D3M) based on the variational principle is proposed for partial differential equations (PDEs). The solution of PDEs can be formulated as the solution of a constrained optimization…
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…
The use of neural networks to approximate partial differential equations (PDEs) has gained significant attention in recent years. However, the approximation of PDEs with localised phenomena, e.g., sharp gradients and singularities, remains…
Neural networks are increasingly used to construct numerical solution methods for partial differential equations. In this expository review, we introduce and contrast three important recent approaches attractive in their simplicity and…
Surface partial differential equations arise in numerous scientific and engineering applications. Their numerical solution on static and evolving surfaces remains challenging due to geometric complexity and, for evolving geometries, the…
Isogeometric analysis is a powerful paradigm which exploits the high smoothness of splines for the numerical solution of high order partial differential equations. However, the tensor-product structure of standard multivariate B-spline…
Using exhaustion method and finite differences a new method to solve system of partial differential equations and is presented. This method allows design algorithm to solve linear and nonlinear systems in irregular domains. Applying this…
We propose a multiscale method for elliptic problems on complex domains, e.g. domains with cracks or complicated boundary. For local singularities this paper also offers a discrete alternative to enrichment techniques such as XFEM. We…
A classic approach for solving differential equations with neural networks builds upon neural forms, which employ the differential equation with a discretisation of the solution domain. Making use of neural forms for time-dependent…
In recent work, Li et al.\ (Comm.\ Math.\ Sci., 7:81-107, 2009) developed a diffuse-domain method (DDM) for solving partial differential equations in complex, dynamic geometries with Dirichlet, Neumann, and Robin boundary conditions. The…
Isogeometric analysis (IGA) is a numerical method that connects computer-aided design (CAD) with finite element analysis (FEA). In CAD the computational domain is usually represented by B-spline or NURBS patches. Given a NURBS…
We present a new approach to three-dimensional electromagnetic scattering problems via fast isogeometric boundary element methods. Starting with an investigation of the theoretical setting around the electric field integral equation within…
In this paper, based on the combination of tensor neural network and a posteriori error estimator, a novel type of machine learning method is proposed to solve high-dimensional boundary value problems with homogeneous and non-homogeneous…