Related papers: Deep Ritz-Finite Element methods: Neural Network M…
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…
The numerical solution of high dimensional partial differential equations (PDEs) is severely constrained by the curse of dimensionality (CoD), rendering classical grid--based methods impractical beyond a few dimensions. In recent years,…
We use elliptic partial differential equations (PDEs) as examples to show various properties and behaviors when shallow neural networks (SNNs) are used to represent the solutions. In particular, we study the numerical ill-conditioning,…
Physics-Informed Neural Networks (PINNs) are mesh-free approaches for the numerical approximation of partial differential equations, where a neural network is trained by minimizing a loss function derived from the governing equations and…
The loss functions of deep neural networks are complex and their geometric properties are not well understood. We show that the optima of these complex loss functions are in fact connected by simple curves over which training and test…
In this work, we present a machine learning approach for reducing the error when numerically solving time-dependent partial differential equations (PDE). We use a fully convolutional LSTM network to exploit the spatiotemporal dynamics of…
We present a weak finite element method for elliptic problems in one space dimension. Our analysis shows that this method has more advantages than the known weak Galerkin method proposed for multi-dimensional problems, for example, it has…
Physics-informed neural networks solve partial differential equations by training neural networks. Since this method approximates infinite-dimensional PDE solutions with finite collocation points, minimizing discretization errors by…
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…
We derive upper bounds on the complexity of ReLU neural networks approximating the solution maps of parametric partial differential equations. In particular, without any knowledge of its concrete shape, we use the inherent…
We investigate the resolution of parabolic PDEs via Extreme Learning Machine (ELMs) Neural Networks, which have a single hidden layer and can be trained at a modest computational cost as compared with Deep Learning Neural Networks. Our…
In this paper, a shallow Ritz-type neural network for solving elliptic equations with delta function singular sources on an interface is developed. There are three novel features in the present work; namely, (i) the delta function…
Conventional finite element methods are known to be tedious in adaptive refinements due to their conformal regularity requirements. Further, the enrichment functions for adaptive refinements are often not readily available in general…
Extreme learning machine (ELM) is a methodology for solving partial differential equations (PDEs) using a single hidden layer feed-forward neural network. It presets the weight/bias coefficients in the hidden layer with random values, which…
This paper discusses a new method to solve definite integrals using artificial neural networks. The objective is to build a neural network that would be a novel alternative to pre-established numerical methods and with the help of a…
In this paper, we consider a time-dependent discrete network model with highly varying connectivity. The approximation by time is performed using an implicit scheme. We propose the coarse scale approximation construction of network models…
We develop a finite element method for the Laplace--Beltrami operator on a surface described by a set of patchwise parametrizations. The patches provide a partition of the surface and each patch is the image by a diffeomorphism of a…
There are many numerical methods for solving partial different equations (PDEs) on manifolds such as classical implicit, finite difference, finite element, and isogeometric analysis methods which aim at improving the interoperability…
The deep energy method (DEM) has been used to solve the elastic deformation of structures with linear elasticity, hyperelasticity, and strain-gradient elasticity material models based on the principle of minimum potential energy. In this…
Physics informed neural networks (PINNs) are deep learning based techniques for solving partial differential equations (PDEs) encounted in computational science and engineering. Guided by data and physical laws, PINNs find a neural network…