35 citations · 209 across the 19 of their papers we have counts for
19 papers
RUNNs: Ritz-Uzawa Neural Networks for Solving Variational Problems
Pablo Herrera, Jamie M. Taylor, Carlos Uriarte +3
Solving Partial Differential Equations (PDEs) using neural networks presents different challenges, including integration errors and spectral bias, often leading to poor approximati…
A Residual Minimization approach for Nonlinear Partial Differential Equations set in Banach spaces
Ignacio Muga, Jorge Perera, Sergio Rojas +1
In this work, we propose and analyze a residual-minimization strategy for the numerical solution of nonlinear PDEs posed in Banach spaces. Given a finite-dimensional trial space an…
Deep Fourier Residual method for solving time-harmonic Maxwell's equations
Jamie M. Taylor, Manuela Bastidas, David Pardo +1
Solving PDEs with machine learning techniques has become a popular alternative to conventional methods. In this context, Neural networks (NNs) are among the most commonly used mach…
Learning quantities of interest from parametric PDEs: An efficient neural-weighted Minimal Residual approach
Ignacio Brevis, Ignacio Muga, David Pardo +2
The efficient approximation of parametric PDEs is of tremendous importance in science and engineering. In this paper, we show how one can train Galerkin discretizations to efficien…
Adaptive stabilized finite elements via residual minimization onto bubble enrichments
José G. Hasbani, Paulina Sepúlveda, Ignacio Muga +2
The Adaptive Stabilized Finite Element method (AS-FEM) developed in Calo et. al. combines the idea of the residual minimization method with the inf-sup stability offered by the dis…
Automatic stabilization of finite-element simulations using neural networks and hierarchical matrices
Tomasz Sluzalec, Mateusz Dobija, Anna Paszynska +2
Petrov-Galerkin formulations with optimal test functions allow for the stabilization of finite element simulations. In particular, given a discrete trial space, the optimal test sp…
A Deep Double Ritz Method (DRM) for solving Partial Differential Equations using Neural Networks
Carlos Uriarte, David Pardo, Ignacio Muga +1
Residual minimization is a widely used technique for solving Partial Differential Equations in variational form. It minimizes the dual norm of the residual, which naturally yields…
A Deep Fourier Residual Method for solving PDEs using Neural Networks
Jamie M. Taylor, David Pardo, Ignacio Muga
When using Neural Networks as trial functions to numerically solve PDEs, a key choice to be made is the loss function to be minimised, which should ideally correspond to a norm of…
An adaptive superconvergent finite element method based on local residual minimization
Ignacio Muga, Sergio Rojas, Patrick Vega
We introduce an adaptive superconvergent finite element method for a class of mixed formulations to solve partial differential equations involving a diffusion term. It combines a s…
Neural Control of Discrete Weak Formulations: Galerkin, Least-Squares and Minimal-Residual Methods with Quasi-Optimal Weights
Ignacio Brevis, Ignacio Muga, Kristoffer G. van der Zee
There is tremendous potential in using neural networks to optimize numerical methods. In this paper, we introduce and analyse a framework for the neural optimization of discrete we…
Isogeometric Residual Minimization Method (iGRM) with Direction Splitting for Non-Stationary Advection-Diffusion Problems
Marcin Los, Judit Munoz-Matute, Ignacio Muga +1
In this paper, we propose a novel computational implicit method, which we call Isogeometric Residual Minimization (iGRM) with direction splitting. The method mixes the benefits res…
Projection in negative norms and the regularization of rough linear functionals
Felipe Millar, Ignacio Muga, Sergio Rojas +1
In order to construct regularizations of continuous linear functionals acting on Sobolev spaces such as , where and is a Lipschitz domain, we propose…
Data-Driven Finite Elements Methods: Machine Learning Acceleration of Goal-Oriented Computations
Ignacio Brevis, Ignacio Muga, Kristoffer G. van der Zee
We introduce the concept of data-driven finite element methods. These are finite-element discretizations of partial differential equations (PDEs) that resolve quantities of interes…
Isogeometric Residual Minimization (iGRM) for Non-Stationary Stokes and Navier-Stokes Problems
Marcin Los, Ignacio Muga, Judit Munoz-Matute +1
We show that it is possible to obtain a linear computational cost FEM-based solver for non-stationary Stokes and Navier-Stokes equations. Our method employs a technique developed b…
An adaptive stabilized conforming finite element method via residual minimization on dual discontinuous Galerkin norms
Victor M. Calo, Alexandre Ern, Ignacio Muga +1
We design and analyze a new adaptive stabilized finite element method. We construct a discrete approximation of the solution in a continuous trial space by minimizing the residual…
Isogeometric Residual Minimization Method (iGRM) with Direction Splitting Preconditoner for Stationary Advection-Diffusion Problems
Victor M Calo, Marcin Łoś, Quanling Deng +2
In this paper, we introduce the isoGeometric Residual Minimization (iGRM) method. The method solves stationary advection-dominated diffusion problems. We stabilize the method via r…
The Discrete-Dual Minimal-Residual Method (DDMRes) for Weak Advection-Reaction Problems in Banach spaces
Ignacio Muga, Matthew J. W. Tyler, Kristoffer G. van der Zee
We propose and analyse a minimal-residual method in discrete dual norms for approximating the solution of the advection-reaction equation in a weak Banach-space setting. The weak f…
Discretization of Linear Problems in Banach Spaces: Residual Minimization, Nonlinear Petrov-Galerkin, and Monotone Mixed Methods
Ignacio Muga, Kristoffer G. van der Zee
This work presents a comprehensive discretization theory for abstract linear operator equations in Banach spaces. The fundamental starting point of the theory is the idea of residu…
Dispersive and dissipative errors in the DPG method with scaled norms for Helmholtz equation
Jay Gopalakrishnan, Ignacio Muga, Nicole Olivares
We consider the discontinuous Petrov-Galerkin (DPG) method, wher the test space is normed by a modified graph norm. The modificatio scales one of the terms in the graph norm by an…