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Research Papers

A Novel General and Robust Method Based on NAOP for Solving Nonlinear Ordinary Differential Equations and Partial Differential Equations by Cellular Neural Networks

[+] Author and Article Information
Jean Chamberlain Chedjou

Assistant Professor
e-mail: jean.chedjou@aau.at

Kyandoghere Kyamakya

Professor
e-mail: kyandoghere.kyamakya@aau.at
Transportation Informatics Group (TIG),
Institute for Smart System Technologies,
Alpen-Adria-Universität, Klagenfurt,
Lakeside Park B04a,
9020 Klagenfurt, Austria

Contributed by the Dynamic Systems Division of ASME for publication in the Journal of Dynamic Systems, Measurement, and Control. Manuscript received July 24, 2011; final manuscript received July 23, 2012; published online March 28, 2013. Assoc. Editor: Bor-Chin Chang.

J. Dyn. Sys., Meas., Control 135(3), 031014 (Mar 28, 2013) (11 pages) Paper No: DS-11-1221; doi: 10.1115/1.4023241 History: Received July 24, 2011; Revised July 23, 2012

This paper develops and validates through a series of presentable examples, a comprehensive high-precision, and ultrafast computing concept for solving nonlinear ordinary differential equations (ODEs) and partial differential equations (PDEs) with cellular neural networks (CNN). The core of this concept is a straightforward scheme that we call "nonlinear adaptive optimization (NAOP),” which is used for a precise template calculation for solving nonlinear ODEs and PDEs through CNN processors. One of the key contributions of this work is to demonstrate the possibility of transforming different types of nonlinearities displayed by various classical and well-known nonlinear equations (e.g., van der Pol-, Rayleigh-, Duffing-, Rössler-, Lorenz-, and Jerk-equations, just to name a few) unto first-order CNN elementary cells, and thereby enabling the easy derivation of corresponding CNN templates. Furthermore, in the case of PDE 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 PDE 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 ultrafast solver of nonlinear ODEs and/or PDEs. This clearly enables a CNN-based, real-time, ultraprecise, and low-cost computational engineering. As proof of concept, two examples of well-known ODEs are considered namely a second-order linear ODE and a second order nonlinear ODE of the van der Pol type. For each of these ODEs, the corresponding precise CNN templates are derived and are used to deduce the expected solutions. An implementation of the concept developed is possible even on embedded digital platforms (e.g., field programmable gate array (FPGA), digital signal processor (DSP), graphics processing unit (GPU), etc.). This opens a broad range of applications. Ongoing works (as outlook) are using NAOP for deriving precise templates for a selected set of practically interesting ODEs and PDEs equation models such as Lorenz-, Rössler-, Navier Stokes-, Schrödinger-, Maxwell-, etc.

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Figures

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Fig. 1

Synoptic representation of the Lagrange function by combining objective function and constraints

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Fig. 2

Illustration of the sign flip (constraints minimization) as a key step toward the convergence of BDMM algorithm. This is achieved through damped oscillations about the constraint subspace.

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Fig. 3

Illustration of the mapping concept of two inputs leading to CNN templates calculation by the novel technique based-NAOP

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Fig. 4

Synoptic representation of the key steps involved in the complete NAOP process leading to the derivation of CNN-templates for stiff ODEs

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Fig. 5

Convergence of the corresponding CNN-templates as achieved by the NAOP process to solve Eq. (15) with ω2=0.850. The values obtained for the CNN-templates are A∧11=1, A∧12=1, A∧21=-0.850, A∧22=1, A11=0, A12=0, A21=0, and A22=0.

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Fig. 6

Convergence of the corresponding CNN-templates as achieved by the NAOP process to solve Eq. (15) with ω2=850. The values of the CNN-templates are A∧11=1, A∧12=1, A∧21=-850, A∧22=1, A11=0, A12=0, A21=0, and A22=0.

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Fig. 7

Benchmarking: Comparison of the results of Eq. (15) obtained from the direct numerical simulation using matlab (column 1) with the results obtained using CNN (column 2). The first line displays the phase portraits (y,y·) versus (x,x·), the second line displays the waveform solutions y(t) versus x(t), and the third line displays the velocities y· versus x·. The initial conditions are y(0)=x(0)=1 and y·(0)=x·(0)=0 and ω2=0.850. The corresponding CNN-templates are A∧11=1, A∧12=1, A∧21=-0.850, A∧22=1, A11=0, A12=0, A21=0, and A22=0.

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Fig. 8

Benchmarking: Comparison of the results of Eq. (15) obtained from the direct numerical simulation using matlab (column 1) with the results obtained using CNN (column 2). The first line displays the phase portraits (y,y·) versus (x,x·), the second line displays the waveform solutions y(t) versus x(t), and the third line displays the velocities y· versus x·. The initial conditions are y(0)=x(0)=1 and y·(0)=x·(0)=0 and ω2=850. The corresponding CNN-templates are A∧11=1, A∧12=1, A∧21=-850, A∧22=1, A11=0, A12=0, A21=0, and A22=0.

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Fig. 9

simulink general graphical representation of the CNN-computing platform for solving second order stiff differential equations (Eq. (16))

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Fig. 10

(a) Convergence of the corresponding state-control CNN templates as achieved by the NOAP process to solve Eq. (16) with ɛ=0.25 and ω=1. The values of the CNN-templates are A∧11=1.0770, A∧12=-0.6300, A∧21=1.3450, and A∧22=0.5850. (b) Convergence of the corresponding feedback CNN templates as achieved by the NOAP process to solve Eq. (16) with ɛ=0.25 and ω=1. The values of the CNN-templates are A11=0.4473, A12=-0.2586, A21=0.00004846, and A22=0.1310.

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Fig. 11

Benchmarking: Comparison of the results of Eq. (16) obtained from the direct numerical simulation using matlab (column 1) with the results obtained using CNN (column 2). The first line displays the phase portraits (y,y·) versus (x,x·), the second line displays the waveform solutions y(t) versus x(t), and the third line displays the velocities y· versus x·. The initial conditions are y(0)=x(0)=1 and y·(0)=x·(0)=0. The parameters are ɛ=0.25 and ω=1. The corresponding CNN-templates are A∧11=1.0770, A∧12=-0.6300, A∧21=1.3450, A∧22=0.5850, A11=0.4473, A12=-0.2586, A21=0.00004846, and A22=0.1310.

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Fig. 12

Benchmarking: Comparison of the results of Eq. (16) obtained from the direct numerical simulation using matlab (column 1) with the results obtained using CNN (column 2). The first line displays the phase portraits (y,y·) versus (x,x·), the second line displays the waveform solutions y(t) versus x(t), and the third line displays the velocities y· versus x·. The initial conditions are y(0)=x(0)=1 and y·(0)=x·(0)=0. The parameters are ɛ=0.5 and ω=1. The corresponding CNN-templates are A∧11=0.8248; A∧12=0.0719; A∧21=-1.1485; A∧22=-0.2428; A11=0.1754; A12=0.9251; A21=0.1480; A22=2.2434.

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Fig. 13

Benchmarking: Comparison of the results of Eq. (16) obtained from the direct numerical simulation using matlab (column 1) with the results obtained using CNN (column 2). The first line displays the phase portraits (y,y·) versus (x,x·), the second line displays the waveform solutions y(t) versus x(t), and the third line displays the velocities y· versus x·. The initial conditions are y(0)=x(0)=1 and y·(0)=x·(0)=0. The parameters are ɛ=1 and ω=1. The corresponding CNN-templates are A∧11=-0.9752; A∧12=0.9223; A∧21=-0.0251; A∧22=0.3350; A11=3.6000; A12=-3.4980; A21=1.3062; A22=0.8800.

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Fig. 14

Benchmarking: Comparison of the results of Eq. (16) obtained from the direct numerical simulation using matlab (column 1) with the results obtained using CNN (column 2). The first line displays the phase portraits (y,y·) versus (x,x·), the second line displays the waveform solutions y(t) versus x(t), and the third line displays the velocities y· versus x·. The initial conditions are y(0)=x(0)=1 and y·(0)=x·(0)=0. The parameters are ɛ=5 and ω=1. The corresponding CNN-templates are A∧11=0.7682; A∧12=0.3890; A∧21=-0.0758; A∧22=0.0614; A11=3.6051; A12=-4.4970; A21=1.3350; A22=0.6512.

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Fig. 15

Benchmarking: Comparison of the results of Eq. (16) obtained from the direct numerical simulation using matlab (column 1) with the results obtained using CNN (column 2). The first line displays the phase portraits (y,y·) versus (x,x·), the second line displays the waveform solutions y(t) versus x(t), and the third line displays the velocities y· versus x·. The initial conditions are y(0)=x(0)=1 and y·(0)=x·(0)=0. The parameters are ɛ=10 and ω=1. The corresponding CNN-templates are A∧11=0.7825; A∧12=0.3890; A∧21=-0.0760; A∧22=0.3300; A11=3.6100; A12=-4.5000; A21=1.3000; A22=0.6806.

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Fig. 16

Global architecture of the computing platform planned to enable a real-time computational engineering. Diverse users may access the CNN processor platforms in a remote way through the Internet.

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Fig. 17

Core idea of the server architecture intended for the CNN based super-computing platform to enable real-time computational engineering. It is a detailed description of the central sever given in Fig. 16.

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