CNN-based ultrafast solver of stiff ODEs and PDEs for enabling realtime Computational Engineering

Purpose - This paper seeks to develop, propose and validate, through a series of presentable examples, a comprehensive high-precision and ultra-fast computing concept for solving stiff ordinary differential equations (ODEs) and partial differential equations (PDEs) with cellular neural networks (CNN...

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Bibliographic Details
Published in:Compel 2011-01, Vol.30 (4), p.1333-1349
Main Authors: Chedjou, J.C., Kyamakya, K.
Format: Article
Language:English
Subjects:
Application programming interface
Architectural engineering
Computation
Differential equations
Electrical engineering
Engineering
Mathematical analysis
Mathematical models
Navier-Stokes equations
Neural networks
Nonlinearity
Ordinary differential equations
Partial differential equations
Rapid prototyping
Solvers
Studies
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Summary:Purpose - This paper seeks to develop, propose and validate, through a series of presentable examples, a comprehensive high-precision and ultra-fast computing concept for solving stiff ordinary differential equations (ODEs) and partial differential equations (PDEs) with cellular neural networks (CNN).Design methodology approach - The core of the concept developed in this paper is a straight-forward scheme that we call "nonlinear adaptive optimization (NAOP)", which is used for a precise template calculation for solving any (stiff) nonlinear ODEs through CNN processors.Findings - One of the key contributions of this work (this is a real breakthrough) is to demonstrate the possibility of mapping transforming different types of nonlinearities displayed by various classical and well-known oscillators (e.g. van der Pol-, Rayleigh-, Duffing-, Rössler-, Lorenz-, and Jerk- oscillators, just to name a few) unto first-order CNN elementary cells, and thereby enabling the easy derivation of corresponding CNN-templates. Furthermore, in case of PDEs solving, the same concept also allows a mapping unto first-order CNN cells while considering one or even more nonlinear terms of the Taylor's series expansion generally used in the transformation of a PDEs in a set of coupled nonlinear ODEs. Therefore, the concept of this paper does significantly contribute to the consolidation of CNN as a universal and ultra-fast solver of stiff differential equations (both ODEs and PDEs). This clearly enables a CNN-based, real-time, ultra-precise, and low-cost Computational Engineering. As proof of concept a well-known prototype of stiff equations (van der Pol) has been considered; the corresponding precise CNN-templates are derived to obtain precise solutions of this equation.Originality value - This paper contributes to the enrichment of the literature as the relevant state-of-the-art does not provide a systematic and robust method to solve nonlinear ODEs and or nonlinear PDEs using the CNN-paradigm. Further, the "NAOP" concept developed in this paper has been proven to perform accurate and robust calculations. This concept is not based on trial-and-error processes as it is the case for various classes of optimization methods tools (e.g. genetic algorithm, particle swarm, neural networks, etc.). The "NAOP" concept developed in this frame does significantly contribute to the consolidation of CNN as a universal and ultra-fast solver of nonlinear differential equations (both ODEs and PDEs).
ISSN:0332-1649
2054-5606
DOI:10.1108/03321641111133235