Related papers: Non-overlapping, Schwarz-type Domain Decomposition…
In this paper, we propose a novel algorithm called Neuron-wise Parallel Subspace Correction Method (NPSC) for the finite neuron method that approximates numerical solutions of partial differential equations (PDEs) using neural network…
We propose a simple domain decomposition method for $d$-dimensional elliptic PDEs which involves an overlapping decomposition into local subdomain problems and a global coarse problem. It relies on a space-filling curve to create equally…
A novel overlapping domain decomposition splitting algorithm based on a Crank-Nisolson method is developed for the stochastic nonlinear Schroedinger equation driven by a multiplicative noise with non-periodic boundary conditions. The…
We present a domain decomposition-based deep learning method for solving elliptic and parabolic interface problems with discontinuous coefficients in two to ten dimensions. Our Multi-Activation Function (MAF) approach employs two…
Extreme learning machines (ELMs), which preset hidden layer parameters and solve for last layer coefficients via a least squares method, can typically solve partial differential equations faster and more accurately than Physics Informed…
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…
Despite the significant success of deep learning in computer vision tasks, cross-domain tasks still present a challenge in which the model's performance will degrade when the training set and the test set follow different distributions.…
Solving partial differential equations (PDEs) is the canonical approach for understanding the behavior of physical systems. However, large scale solutions of PDEs using state of the art discretization techniques remains an expensive…
We introduce adaptive-basis physics-informed neural networks (AB-PINNs), a novel approach to domain decomposition for training PINNs in which existing subdomains dynamically adapt to the intrinsic features of the unknown solution. Drawing…
Many important problems in science and engineering require solving the so-called parametric partial differential equations (PDEs), i.e., PDEs with different physical parameters, boundary conditions, shapes of computation domains, etc.…
Domain decomposition (DD) methods for solving time-dependent problems can be classified by (i) the method of domain decomposition used, (ii) the choice of decomposition operators (exchange of boundary conditions), and (iii) the splitting…
We consider solving complex spatiotemporal dynamical systems governed by partial differential equations (PDEs) using frequency domain-based discrete learning approaches, such as Fourier neural operators. Despite their widespread use for…
In this paper we are concerned with restricted additive Schwarz with local impedance transformation conditions for a family of Helmholtz problems in two dimensions. These problems are discretized by the finite element method with conforming…
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…
The physics informed neural network (PINN) is a promising method for solving time-evolution partial differential equations (PDEs). However, the standard PINN method may fail to solve the PDEs with strongly nonlinear characteristics or those…
There have been growing interests in leveraging experimental measurements to discover the underlying partial differential equations (PDEs) that govern complex physical phenomena. Although past research attempts have achieved great success…
Barren plateaus present a major challenge in the training of variational quantum algorithms (VQAs), particularly for large-scale discretizations of nonlinear partial differential equations. In this work, we introduce a domain decomposition…
Deep learning has gained attention for solving PDEs, but the black-box nature of neural networks hinders precise enforcement of boundary conditions. To address this, we propose a boundary condition-guaranteed evolutionary Kolmogorov-Arnold…
We present a Schur complement Domain Decomposition (DD) algorithm for the solution of frequency domain multiple scattering problems. Just as in the classical DD methods we (1) enclose the ensemble of scatterers in a domain bounded by an…
Modelling the underlying mechanisms of neurodegenerative diseases demands methods that capture heterogeneous and spatially varying dynamics from sparse, high-dimensional neuroimaging data. Integrating partial differential equation (PDE)…