Related papers: Extended Galerkin neural network approximation of …

Nonlinear optimal control problems in Hilbert spaces are considered for which we derive approximation theorems for Galerkin approximations. Approximation theorems are available in the literature. The originality of our approach relies on…

Despite its long history, Bayesian neural networks (BNNs) and variational training remain underused in practice: standard Gaussian posteriors misalign with network geometry, KL terms can be brittle in high dimensions, and implementations…

Hybrid Bayesian Networks (HBNs), which contain both discrete and continuous variables, arise naturally in many application areas (e.g., image understanding, data fusion, medical diagnosis, fraud detection). This paper concerns inference in…

Discontinuous Galerkin (DG) methods for the numerical solution of partial differential equations have enjoyed considerable success because they are both flexible and robust: They allow arbitrary unstructured geometries and easy control of…

Cartoon-texture image decomposition is a critical preprocessing problem bottlenecked by the numerical intractability of classical variational or optimization models and the tedious manual tuning of global regularization parameters.We…

Learning to optimize is a rapidly growing area that aims to solve optimization problems or improve existing optimization algorithms using machine learning (ML). In particular, the graph neural network (GNN) is considered a suitable ML model…

This note explores the applicability of unsupervised machine learning techniques towards hard optimization problems on random inputs. In particular we consider Graph Neural Networks (GNNs) -- a class of neural networks designed to learn…

In this paper, we present a staggered discontinuous Galerkin (SDG) method for a class of nonlinear elliptic equations in two dimensions. The SDG methods have some distinctive advantages, and have been successfully applied to a wide range of…

In this work, we are concerned with neural network guided goal-oriented a posteriori error estimation and adaptivity using the dual weighted residual method. The primal problem is solved using classical Galerkin finite elements. The adjoint…

Bayesian Neural Networks (BNNs) provide a tool to estimate the uncertainty of a neural network by considering a distribution over weights and sampling different models for each input. In this paper, we propose a method for uncertainty…

Variational approximations to Gaussian processes (GPs) typically use a small set of inducing points to form a low-rank approximation to the covariance matrix. In this work, we instead exploit a sparse approximation of the precision matrix.…

Deep Neural Networks (DNNs) are widely used for their ability to effectively approximate large classes of functions. This flexibility, however, makes the strict enforcement of constraints on DNNs an open problem. Here we present a framework…

Graph processes exhibit a temporal structure determined by the sequence index and and a spatial structure determined by the graph support. To learn from graph processes, an information processing architecture must then be able to exploit…

One of the core problems in variational inference is a choice of approximate posterior distribution. It is crucial to trade-off between efficient inference with simple families as mean-field models and accuracy of inference. We propose a…

Machine learning methods have been lately used to solve partial differential equations (PDEs) and dynamical systems. These approaches have been developed into a novel research field known as scientific machine learning in which techniques…

We perform a posteriori error analysis in the supremum norm for the quadratic discontinuous Galerkin method for the elliptic obstacle problem. We define two discrete sets (motivated by Gaddam, Gudi and Kamana [1]), one set having integral…

We propose an augmented Lagrangian-type algorithm for the solution of generalized Nash equilibrium problems (GNEPs). Specifically, we discuss the convergence properties with regard to both feasibility and optimality of limit points. This is…

In this paper, we present a new variational integrator for problems in Lagrangian mechanics. Using techniques from Galerkin variational integrators, we construct a scheme for numerical integration that converges geometrically, and is…

Geometrically parametrized Partial Differential Equations are nowadays widely used in many different fields as, for example, shape optimization processes or patient specific surgery studies. The focus of this work is on some advances for…

Generative neural networks are able to mimic intricate probability distributions such as those of handwritten text, natural images, etc. Since their inception several models were proposed. The most successful of these were based on…