Neural network algorithm based on Legendre improved extreme learning machine for solving elliptic partial differential equations

2019 ◽  
Vol 24 (2) ◽  
pp. 1083-1096 ◽  
Author(s):  
Yunlei Yang ◽  
Muzhou Hou ◽  
Hongli Sun ◽  
Tianle Zhang ◽  
Futian Weng ◽  
...  
Keyword(s):  
Neural Network ◽  
Partial Differential Equations ◽  
Differential Equations ◽  
Extreme Learning Machine ◽  
Elliptic Partial Differential Equations ◽  
Partial Differential ◽  
Network Algorithm ◽  
Neural Network Algorithm ◽  
Learning Machine

scholarly journals A Deep Neural Network Algorithm for Semilinear Elliptic PDEs with Applications in Insurance Mathematics

Risks ◽  
2020 ◽  
Vol 8 (4) ◽  
pp. 136
Author(s):  
Stefan Kremsner ◽  
Alexander Steinicke ◽  
Michaela Szölgyenyi
Keyword(s):  
Neural Network ◽  
Partial Differential Equations ◽  
Differential Equations ◽  
Deep Neural Network ◽  
Risk Measure ◽  
Elliptic Partial Differential Equations ◽  
High Dimensional ◽  
Control Problems ◽  
Partial Differential ◽  
Neural Network Algorithm

In insurance mathematics, optimal control problems over an infinite time horizon arise when computing risk measures. An example of such a risk measure is the expected discounted future dividend payments. In models which take multiple economic factors into account, this problem is high-dimensional. The solutions to such control problems correspond to solutions of deterministic semilinear (degenerate) elliptic partial differential equations. In the present paper we propose a novel deep neural network algorithm for solving such partial differential equations in high dimensions in order to be able to compute the proposed risk measure in a complex high-dimensional economic environment. The method is based on the correspondence of elliptic partial differential equations to backward stochastic differential equations with unbounded random terminal time. In particular, backward stochastic differential equations—which can be identified with solutions of elliptic partial differential equations—are approximated by means of deep neural networks.

A normal equation-based extreme learning machine for solving linear partial differential equations

2021 ◽  
pp. 1-12
Author(s):  
Vikas Dwivedi ◽  
Balaji Srinivasan
Keyword(s):  
Neural Network ◽  
Partial Differential Equations ◽  
Differential Equations ◽  
Extreme Learning Machine ◽  
Normal Equation ◽  
Partial Differential ◽  
Normal Equations ◽  
Linear Partial Differential Equations ◽  
Learning Machine ◽  
Complicated Geometries

Abstract This paper develops an extreme learning machine for solving linear partial differential equations (PDE) by extending the normal equations approach for linear regression. The normal equations method is typically used when the amount of available data is small. In PDEs, the only available ground truths are the boundary and initial conditions (BC and IC). We use the physics-based cost function used in state-of-the-art deep neural network-based PDE solvers called physics informed neural network (PINN) to compensate for the small data. However, unlike PINN, we derive the normal equations for PDEs and directly solve them to compute the network parameters. We demonstrate our method's feasibility and efficiency by solving several problems like function approximation, solving ordinary differential equations (ODEs), steady and unsteady PDEs on regular and complicated geometries. We also highlight our method's limitation in capturing sharp gradients and propose its domain distributed version to overcome this issue. We show that this approach is much faster than traditional gradient descent-based approaches and offers an alternative to conventional numerical methods in solving PDEs in complicated geometries.

The Adaptive Ensemble of OP-ELM Using Forward-Backward Selection

2013 ◽  
Vol 427-429 ◽  
pp. 1666-1669
Author(s):  
Bo Han ◽  
Bo He ◽  
Meng Meng Ma
Keyword(s):  
Neural Network ◽  
Extreme Learning Machine ◽  
Selection Algorithm ◽  
New Approach ◽  
Adaptive Weights ◽  
Network Algorithm ◽  
Neural Network Algorithm ◽  
Learning Machine

Extreme learning machine (ELM) as a neural network algorithm has shown its good performance in regression or classification applications, but it has a weak robustness. In this paper, a new approach called The Adaptive Ensemble Of OP-ELM using Forward-Backward Selection (AEOP-ELM) is presented, it consists of two significant steps, firstly, we use forward-Backward selection algorithm to select the inputs which will ensure the robustness of the output, then, we train several independent OP-ELM models, and we test them iteratively to find the adaptive weights which will improve the accuracy of the output. The experiments indicate the AEOP-ELM achieves a better robustness than the original ELM as well as a better accuracy.

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