Related papers: Multigrid-Augmented Deep Learning Preconditioners …
A new method to solve computationally challenging (random) parametric obstacle problems is developed and analyzed, where the parameters can influence the related partial differential equation (PDE) and determine the position and surface…
Three-dimensional target identification using scattering techniques requires high accuracy solutions and very fast computations for real-time predictions in some critical applications. We first train a deep neural operator~(DeepONet) to…
We present a novel architecture for learning geometry-aware preconditioners for linear partial differential equations (PDEs). We show that a deep operator network (Deeponet) can be trained on a simple geometry and remain a robust…
In this paper we are concerned with fast algorithms for the systems arising from the plane wave discretizations for two-dimensional Helmholtz equations with large wave numbers. We consider the plane wave weighted least squares (PWLS) method…
Models with dominant advection always posed a difficult challenge for projection-based reduced order modelling. Many methodologies that have recently been proposed are based on the pre-processing of the full-order solutions to accelerate…
Recently deep learning-based methods have been applied in image compression and achieved many promising results. In this paper, we propose an improved hybrid layered image compression framework by combining deep learning and the traditional…
This paper studies and analyzes a preconditioned Krylov solver for Helmholtz problems that are formulated with absorbing boundary layers based on complex coordinate stretching. The preconditioner problem is a Helmholtz problem where not…
This paper proposes a novel deep subspace clustering approach which uses convolutional autoencoders to transform input images into new representations lying on a union of linear subspaces. The first contribution of our work is to insert…
Large linear systems are ubiquitous in modern computational science and engineering. The main recipe for solving them is the use of Krylov subspace iterative methods with well-designed preconditioners. Recently, GNNs have been shown to be a…
The Helmholtz equation is related to seismic exploration, sonar, antennas, and medical imaging applications. It is one of the most challenging problems to solve in terms of accuracy and convergence due to the scalability issues of the…
We present a novel differentiable grid-based representation for efficiently solving differential equations (DEs). Widely used architectures for neural solvers, such as sinusoidal neural networks, are coordinate-based MLPs that are both…
Even though convolutional neural networks have become the method of choice in many fields of computer vision, they still lack interpretability and are usually designed manually in a cumbersome trial-and-error process. This paper aims at…
In this work we extend the shifted Laplacian approach to the elastic Helmholtz equation. The shifted Laplacian multigrid method is a common preconditioning approach for the discretized acoustic Helmholtz equation. In some cases, like…
We introduce a neural-preconditioned iterative solver for Poisson equations with mixed boundary conditions. Typical Poisson discretizations yield large, ill-conditioned linear systems. Iterative solvers can be effective for these problems,…
The paper introduces the sweeping preconditioner, which is highly efficient for iterative solutions of the variable coefficient Helmholtz equation including very high frequency problems. The first central idea of this novel approach is to…
Multigrid preconditioners and solvers for the indefinite Helmholtz equation suffer from non-stability of the stationary smoothers due to the indefinite spectrum of the operator. In this paper we explore GMRES as a replacement for the…
Getting standard multigrid to work efficiently for the high-frequency Helmholtz equation has been an open problem in applied mathematics for years. Much effort has been dedicated to finding solution methods which can use multigrid…
We develop a deep learning approach to extract ray directions at discrete locations by analyzing highly oscillatory wave fields. A deep neural network is trained on a set of local plane-wave fields to predict ray directions at discrete…
The resolution of the Poisson equation is usually one of the most computationally intensive steps for incompressible fluid solvers. Lately, Deep Learning, and especially Convolutional Neural Networks (CNN), has been introduced to solve this…
Convolutional neural networks (CNNs) have shown great capability of solving various artificial intelligence tasks. However, the increasing model size has raised challenges in employing them in resource-limited applications. In this work, we…