Related papers: DPG loss functions for learning parameter-to-solut…
Many partial differential equations (PDEs) such as Navier--Stokes equations in fluid mechanics, inelastic deformation in solids, and transient parabolic and hyperbolic equations do not have an exact, primal variational structure. Recently,…
Learning mappings between functional spaces, also known as function-on-function regression, is a fundamental problem in functional data analysis with broad applications, including spatiotemporal forecasting, curve prediction, and climate…
Machine learning for scientific applications faces the challenge of limited data. We propose a framework that leverages a priori known physics to reduce overfitting when training on relatively small datasets. A deep neural network is…
Traditional, numerical discretization-based solvers of partial differential equations (PDEs) are fundamentally agnostic to domains, boundary conditions and coefficients. In contrast, machine learnt solvers have a limited generalizability…
In this paper, we consider approximating the parameter-to-solution maps of parametric partial differential equations (PPDEs) using deep neural networks (DNNs). We propose an efficient approach combining reduced collocation methods (RCMs)…
We develop and analyze an ultraweak formulation of linear PDEs in nondivergence form where the coefficients satisfy the Cordes condition. Based on the ultraweak formulation we propose discontinuous Petrov--Galerkin (DPG) methods. We…
The construction of loss functions presents a major challenge in data-driven modeling involving weak-form operators in PDEs and gradient flows, particularly due to the need to select test functions appropriately. We address this challenge…
This paper analyzes the convergence rate of a deep Galerkin method for the weak solution (DGMW) of second-order elliptic partial differential equations on $\mathbb{R}^d$ with Dirichlet, Neumann, and Robin boundary conditions, respectively.…
It is expensive to compute residual diffusivity in chaotic in-compressible flows by solving advection-diffusion equation due to the formation of sharp internal layers in the advection dominated regime. Proper orthogonal decomposition (POD)…
We introduce two-scale loss functions for use in various gradient descent algorithms applied to classification problems via deep neural networks. This new method is generic in the sense that it can be applied to a wide range of machine…
Metalearning of deep neural network (DNN) architectures and hyperparameters has become an increasingly important area of research. Loss functions are a type of metaknowledge that is crucial to effective training of DNNs, however, their…
Differential equations are used in a wide variety of disciplines, describing the complex behavior of the physical world. Analytic solutions to these equations are often difficult to solve for, limiting our current ability to solve complex…
Super-resolution using deep neural networks typically relies on highly curated training sets that are often unavailable in clinical deployment scenarios. Using loss functions that assume Gaussian-distributed residuals makes the learning…
Recent research has used deep learning to develop partial differential equation (PDE) models in science and engineering. The functional form of the PDE is determined by a neural network, and the neural network parameters are calibrated to…
Solving parametric Partial Differential Equations (PDEs) for a broad range of parameters is a critical challenge in scientific computing. To this end, neural operators, which \textcolor{black}{predicts the PDE solution with variable PDE…
We analyze the discontinuous Petrov-Galerkin (DPG) method with optimal test functions when applied to solve the Reissner-Mindlin model of plate bending. We prove that the hybrid variational formulation underlying the DPG method is…
We analyze a novel multi-level version of a recently introduced compressed sensing (CS) Petrov-Galerkin (PG) method from [H. Rauhut and Ch. Schwab: Compressive Sensing Petrov-Galerkin approximation of high-dimensional parametric operator…
We present a model and variance reduction method for the fast and reliable computation of statistical outputs of stochastic elliptic partial differential equations. Our method consists of three main ingredients: (1) the hybridizable…
This paper constitutes our initial effort in developing sparse grid discontinuous Galerkin (DG) methods for high-dimensional partial differential equations (PDEs). Over the past few decades, DG methods have gained popularity in many…
The aim of this work is to consider multiscale algorithms for solving PDEs with Galerkin methods on bounded domains. We provide results on convergence and condition numbers. We show how to handle PDEs with Dirichlet boundary conditions. We…