Related papers: Solving High-Dimensional PDEs Using Linearized Neu…
Linear programming has played a crucial role in shaping decision-making, resource allocation, and cost reduction in various domains. In this paper, we investigate the application of overparametrized neural networks and their implicit bias…
Rectified Linear Units (ReLU) have become the main model for the neural units in current deep learning systems. This choice has been originally suggested as a way to compensate for the so called vanishing gradient problem which can undercut…
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…
Randomized neural network (RaNN) methods have been proposed for solving various partial differential equations (PDEs), demonstrating high accuracy and efficiency. However, initializing the fixed parameters remains challenging. Additionally,…
Physics-informed neural networks (PINNs) [31] use automatic differentiation to solve partial differential equations (PDEs) by penalizing the PDE in the loss function at a random set of points in the domain of interest. Here, we develop a…
This paper proposes a rank inspired neural network (RINN) to tackle the initialization sensitivity issue of physics informed extreme learning machines (PIELM) when numerically solving partial differential equations (PDEs). Unlike PIELM…
We develop the Randomized Neural Networks with Petrov-Galerkin Methods (RNN-PG methods) to solve linear elasticity problems. RNN-PG methods use Petrov-Galerkin variational framework, where the solution is approximated by randomized neural…
The extreme learning machine (ELM) method can yield highly accurate solutions to linear/nonlinear partial differential equations (PDEs), but requires the last hidden layer of the neural network to be wide to achieve a high accuracy. If the…
In this paper, we propose and study neural network based methods for solutions of high-dimensional quadratic porous medium equation (QPME). Three variational formulations of this nonlinear PDE are presented: a strong formulation and two…
Many Partial Differential Equations (PDEs) do not have analytical solution, and can only be solved by numerical methods. In this context, Physics-Informed Neural Networks (PINN) have become important in the last decades, since it uses a…
We combine concepts from multilevel solvers for partial differential equations (PDEs) with neural network based deep learning and propose a new methodology for the efficient numerical solution of high-dimensional parametric PDEs. An…
Successfully training deep neural networks often requires either batch normalization, appropriate weight initialization, both of which come with their own challenges. We propose an alternative, geometrically motivated method for training.…
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…
Neural networks with randomly generated hidden weights (RaNNs) have been extensively studied, both as a standalone learning method and as an initialization for fully trainable deep learning methods. In this work, we study RaNN expressivity…
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 consider training over-parameterized two-layer neural networks with Rectified Linear Unit (ReLU) using gradient descent (GD) method. Inspired by a recent line of work, we study the evolutions of network prediction errors across GD…
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…
We draw connections between simple neural networks and under-determined linear systems to comprehensively explore several interesting theoretical questions in the study of neural networks. First, we emphatically show that it is unsurprising…
In this paper we propose a method to generate suitably refined finite element meshes using neural networks. As a model problem we consider a linear elasticity problem on a planar domain (possibly with holes) having a polygonal boundary. We…
We introduce a new numerical method based on machine learning to approximate the solution of elliptic partial differential equations with collocation using a set of sigmoidal functions. We show that a feedforward neural network with a…