Related papers: Galerkin Neural Networks: A Framework for Approxim…
Parametric partial differential equations (PDEs) are fundamental for modeling a wide range of physical and engineering systems influenced by uncertain or varying parameters. Traditional neural network-based solvers, such as Physics-Informed…
We present a new type of modified Galerkin method. It is a construction with several (inductively defined) levels, that provides approximate solutions of increasing accuracy with every new level. These solutions are constructed as…
Discontinuous Galerkin (DG) discretizations with exact representation of the geometry and local polynomial degree adaptivity are revisited. Hybridization techniques are employed to reduce the computational cost of DG approximations and…
The weak Galerkin (WG) finite element method is an effective and flexible general numerical technique for solving partial differential equations. It is a natural extension of the classic conforming finite element method for discontinuous…
A framework for performing dynamic mesh adaptation with the discontinuous Galerkin method (DGM) is presented. Adaptations include modifications of the local mesh step size (h-adaptation) and the local degree of the approximating polynomials…
Variational representations of divergences and distances between high-dimensional probability distributions offer significant theoretical insights and practical advantages in numerous research areas. Recently, they have gained popularity in…
In this work we apply the Deep Galerkin Method (DGM) described in Sirignano and Spiliopoulos (2018) to solve a number of partial differential equations that arise in quantitative finance applications including option pricing, optimal…
In this paper, we use Fourier analysis to study the superconvergence of the semi-discrete discontinuous Galerkin method for scalar linear advection equations in one spatial dimension. The error bounds and asymptotic errors are derived for…
These notes present preliminary results regarding two different approximations of linear infinite-horizon optimal control problems arising in model predictive control. Input and state trajectories are parametrized with basis functions and a…
The paper deals with a stochastic Galerkin approximation of elliptic Dirichlet boundary control problems with random input data. The expectation of a tracking cost functional with the deterministic constrained control is minimized. Error…
We introduce in this paper a technique for the reduced order approximation of parametric symmetric elliptic partial differential equations. For any given dimension, we prove the existence of an optimal subspace of at most that dimension…
We propose a projection-based model order reduction method for the solution of parameter-dependent dynamical systems. The proposed method relies on the construction of time-dependent reduced spaces generated from evaluations of the solution…
This paper develops a unified general framework for designing convergent finite difference and discontinuous Galerkin methods for approximating viscosity and regular solutions of fully nonlinear second order PDEs. Unlike the well-known…
We present a novel deep learning approach to approximate the solution of large, sparse, symmetric, positive-definite linear systems of equations. These systems arise from many problems in applied science, e.g., in numerical methods for…
In this work, we consider the approximation of parametric maps using the so-called Galerkin POD-NN method. This technique combines the computation of a reduced basis via proper orthogonal decomposition (POD) and artificial neural networks…
In this paper we develop an $hp$-adaptive procedure for the numerical solution of general, semilinear elliptic boundary value problems in 1d, with possible singular perturbations. Our approach combines both a prediction-type adaptive Newton…
In this paper, we concentrate on the superconvergence of the local discontinuous Galerkin method with generalized numerical fluxes for one-dimensional linear time-dependent fourth-order equations. The adjustable numerical viscosity of the…
This work considers the Galerkin approximation and analysis for a hyperbolic integrodifferential equation, where the non-positive variable-sign kernel and nonlinear-nonlocal damping with both the weak and viscous damping effects are…
The elucidation of many physical problems in science and engineering is subject to the accurate numerical modelling of complex wave propagation phenomena. Over the last decades, high-order numerical approximation for partial differential…
To solve high-dimensional parameter-dependent partial differential equations (pPDEs), a neural network architecture is presented. It is constructed to map parameters of the model data to corresponding finite element solutions. To improve…