Numerical Computation of Partial Differential Equations by Hidden-Layer Concatenated Extreme Learning Machine

04/24/2022
by   Naxian Ni, et al.
0

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 last hidden layer is narrow, the accuracy of the existing ELM method will be poor, irrespective of the rest of the network configuration. In this paper we present a modified ELM method, termed HLConcELM (hidden-layer concatenated ELM), to overcome the above drawback of the conventional ELM method. The HLConcELM method can produce highly accurate solutions to linear/nonlinear PDEs when the last hidden layer of the network is narrow and when it is wide. The new method is based on a type of modified feedforward neural networks (FNN), termed HLConcFNN (hidden-layer concatenated FNN), which incorporates a logical concatenation of the hidden layers in the network and exposes all the hidden nodes to the output-layer nodes. HLConcFNNs have the interesting property that, given a network architecture, when additional hidden layers are appended to the network or when extra nodes are added to the existing hidden layers the representation capacity of the HLConcFNN associated with the new architecture is guaranteed to be not smaller than that of the original network architecture. Here representation capacity refers to the set of all functions that can be exactly represented by the neural network of a given architecture. We present ample benchmark tests with linear/nonlinear PDEs to demonstrate the computational accuracy and performance of the HLConcELM method and the superiority of this method to the conventional ELM from previous works.

READ FULL TEXT

page 5

page 13

page 16

page 20

page 23

page 27

page 28

research
09/13/2023

An Extreme Learning Machine-Based Method for Computational PDEs in Higher Dimensions

We present two effective methods for solving high-dimensional partial di...
research
12/04/2020

Local Extreme Learning Machines and Domain Decomposition for Solving Linear and Nonlinear Partial Differential Equations

We present a neural network-based method for solving linear and nonlinea...
research
03/18/2021

Linear Iterative Feature Embedding: An Ensemble Framework for Interpretable Model

A new ensemble framework for interpretable model called Linear Iterative...
research
09/20/2021

Machine-learning hidden symmetries

We present an automated method for finding hidden symmetries, defined as...
research
02/01/2018

A Modified Sigma-Pi-Sigma Neural Network with Adaptive Choice of Multinomials

Sigma-Pi-Sigma neural networks (SPSNNs) as a kind of high-order neural n...
research
06/29/2018

Neural Networks Trained to Solve Differential Equations Learn General Representations

We introduce a technique based on the singular vector canonical correlat...

Please sign up or login with your details

Forgot password? Click here to reset