Related papers: Finite element interpolated neural networks for so…

Neural networks have become a prominent approach to solve inverse problems in recent years. While a plethora of such methods was developed to solve inverse problems empirically, we are still lacking clear theoretical guarantees for these…

Neural network systems describe complex mappings that can be very difficult to understand. In this paper, we study the inverse problem of determining the input images that get mapped to specific neural network classes. Ultimately, we expect…

We propose a new framework for processing Fringe Patterns (FP). Our novel approach builds upon the hypothesis that the denoising and normalisation of FPs can be learned by a deep neural network if enough pairs of corrupted and ideal FPs are…

Inverse problems arise in a number of domains such as medical imaging, remote sensing, and many more, relying on the use of advanced signal and image processing approaches -- such as sparsity-driven techniques -- to determine their…

Inverse problems arise anywhere we have indirect measurement. As, in general they are ill-posed, to obtain satisfactory solutions for them needs prior knowledge. Classically, different regularization methods and Bayesian inference based…

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…

We present a methodology combining neural networks with physical principle constraints in the form of partial differential equations (PDEs). The approach allows to train neural networks while respecting the PDEs as a strong constraint in…

We explore the use of graph neural networks (GNNs) to model spatial processes in which there is no a priori graphical structure. Similar to finite element analysis, we assign nodes of a GNN to spatial locations and use a computational…

Neural network-based approaches for solving partial differential equations (PDEs) have recently received special attention. However, the large majority of neural PDE solvers only apply to rectilinear domains, and do not systematically…

This paper presents a new data-driven finite element framework that is applicable to a broad range of engineering simulation problems. In the data-driven approach, the conservation laws and boundary conditions are satisfied by means of the…

Despite -- or maybe because of -- their astonishing capacity to fit data, neural networks are believed to have difficulties extrapolating beyond training data distribution. This work shows that, for extrapolations based on finite…

In this work, we present a study combining two approaches in the context of solving PDEs: the continuous finite element method (FEM) and more recent techniques based on neural networks. In recent years, physics-informed neural networks…

Downward continuation is a critical task in potential field processing, including gravity and magnetic fields, which aims to transfer data from one observation surface to another that is closer to the source of the field. Its effectiveness…

In this paper we design a neural interpolation operator to improve the boundary data for regional weather models, which is a challenging problem as we are required to map multi-scale dynamics between grid resolutions. In particular, we…

In a general setting, we study a posteriori estimates used in finite element analysis to measure the error between a solution and its approximation. The latter is not necessarily generated by a finite element method. We show that the error…

Deep neural networks, despite their success in numerous applications, often function without established theoretical foundations. In this paper, we bridge this gap by drawing parallels between deep learning and classical numerical analysis.…

Coupled multiphysics simulations for high-dimensional, large-scale problems can be prohibitively expensive due to their computational demands. This article presents a novel framework integrating a deep operator network (DeepONet) with the…

We present an end-to-end framework for generating solutions to combinatorial optimization problems with unknown components using transformer-based sequence-to-sequence neural networks. Our framework learns directly from past solutions and…

In this paper we propose a new method to stabilise non-symmetric indefinite problems. The idea is to solve a forward and an adjoint problem simultaneously using a suitable stabilised finite element method. Both stabilisation of the element…

Large sparse symmetric linear systems appear in several branches of science and engineering thanks to the widespread use of the finite element method (FEM). The fastest sparse linear solvers available implement hybrid iterative methods.…