Related papers: Error analysis for hybrid finite element/neural ne…
We show that an optimal finite element mesh refinement algorithm for a prototypical elliptic PDE can be learned by a recurrent neural network with a fixed number of trainable parameters independent of the desired accuracy and the input…
The splitting method is a powerful method for solving partial differential equations. Various splitting methods have been designed to separate different physics, nonlinearities, and so on. Recently, a new splitting approach has been…
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…
The paper presents the first rigorous error analysis of an unfitted finite element method for a linear parabolic problem posed on an evolving domain $\Omega(t)$ that may undergo a topological change, such as, for example, a domain…
This paper develops a high-accuracy algorithm for time fractional wave problems, which employs a spectral method in the temporal discretization and a finite element method in the spatial discretization. Moreover, stability and convergence…
The numerical simulation of complex physical processes requires the use of economical discrete models. This lecture presents a general paradigm of deriving a posteriori error estimates for the Galerkin finite element approximation of…
We consider finite element approximations of ill-posed elliptic problems with conditional stability. The notion of {\emph{optimal error estimates}} is defined including both convergence with respect to mesh parameter and perturbations in…
We analyze adaptive mesh-refining algorithms for conforming finite element discretizations of certain non-linear second-order partial differential equations. We allow continuous polynomials of arbitrary, but fixed polynomial order. The…
In the context of unfitted finite element discretizations the realization of high order methods is challenging due to the fact that the geometry approximation has to be sufficiently accurate. Recently a new unfitted finite element method…
In this paper, an Artificial Neural Network (ANN) technique is developed to find solution of celebrated Fractional order Differential Equations (FDE). Compared to integer order differential equation, FDE has the advantage that it can better…
Classic algorithms and machine learning systems like neural networks are both abundant in everyday life. While classic computer science algorithms are suitable for precise execution of exactly defined tasks such as finding the shortest path…
We propose an analysis for the stabilized finite element methods proposed in, E. Burman, Stabilized finite element methods for nonsymmetric, noncoercive, and ill-posed problems. Part I: Elliptic equations. SIAM J. Sci. Comput., 35(6) 2013,…
A feed-forward neural network has a remarkable property which allows the network itself to be a universal approximator for any functions.Here we present a universal, machine-learning based solver for multi-variable partial differential…
This work introduces and analyzes a finite element scheme for evolution problems involving fractional-in-time and in-space differentiation operators up to order two. The left-sided fractional-order derivative in time we consider is employed…
When modeling scientific and industrial problems, geometries are typically modeled by explicit boundary representations obtained from computer-aided design software. Unfitted (also known as embedded or immersed) finite element methods offer…
Nonlinear differential equations are challenging to solve numerically and are important to understanding the dynamics of many physical systems. Deep neural networks have been applied to help alleviate the computational cost that is…
This paper presents machine learning techniques and deep reinforcement learningbased algorithms for the efficient resolution of nonlinear partial differential equations and dynamic optimization problems arising in investment decisions and…
Physics Informed Neural Networks is a numerical method which uses neural networks to approximate solutions of partial differential equations. It has received a lot of attention and is currently used in numerous physical and engineering…
To address the challenge of segmenting noisy images with blurred or fragmented boundaries, this paper presents a robust version of Variational Model Based Tailored UNet (VM_TUNet), a hybrid framework that integrates variational methods with…
The accurate numerical solution of partial differential equations is a central task in numerical analysis allowing to model a wide range of natural phenomena by employing specialized solvers depending on the scenario of application. Here,…