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Research Papers

Constructing Multilayer Feedforward Neural Networks to Approximate Nonlinear Functions in Engineering Mechanics Applications

[+] Author and Article Information
Jin-Song Pei

School of Civil Engineering and Environmental Science, University of Oklahoma, Norman, OK 73019-1024

Eric C. Mai

School of Civil Engineering and Environmental Science, Honors College, University of Oklahoma, Norman, OK 73019-1024

J. Appl. Mech 75(6), 061002 (Aug 15, 2008) (12 pages) doi:10.1115/1.2957600 History: Received May 26, 2006; Revised August 26, 2007; Published August 15, 2008

This paper presents a major step in the development and validation of a systematic prototype-based methodology for designing multilayer feedforward neural networks to model nonlinearities common in engineering mechanics. The applications of this work include (but are not limited to) system identification of nonlinear dynamic systems and neural-network-based damage detection. In this and previous studies (Pei, J. S., 2001, “Parametric and Nonparametric Identification of Nonlinear Systems,” Ph.D. thesis, Columbia University; Pei, J. S., and Smyth, A. W., 2006, “A New Approach to Design Multilayer Feedforward Neural Network Architecture in Modeling Nonlinear Restoring Forces. Part I: Formulation  ,” J. Eng. Mech., 132(12), pp. 1290–1300; Pei, J. S., and Smyth, A. W., 2006, “A New Approach to Design Multilayer Feedforward Neural Network Architecture in Modeling Nonlinear Restoring Forces. Part II: Applications  ,” J. Eng. Mech., 132(12), pp. 1301–1312; Pei, J. S., Wright, J. P., and Smyth, A. W., 2005, “Mapping Polynomial Fitting Into Feedforward Neural Networks for Modeling Nonlinear Dynamic Systems and Beyond  ,” Comput. Methods Appl. Mech. Eng., 194(42–44), pp. 4481–4505), the authors do not presume to provide a universal method to approximate any arbitrary function. Rather the focus is given to the development of a procedure which will consistently lead to successful approximations of nonlinear functions within the specified field. This is done by examining the dominant features of the function to be approximated and exploiting the strength of the sigmoidal basis function. As a result, a greater efficiency and understanding of both neural network architecture (e.g., the number of hidden nodes) as well as weight and bias values is achieved. Through the use of illuminating mathematical insights and a large number of training examples, this study demonstrates the simplicity, power, and versatility of the proposed prototype-based initialization methodology. A clear procedure for initializing neural networks to model various nonlinear functions commonly seen in engineering mechanics is provided. The proposed methodology is compared with the widely used Nguyen–Widrow initialization to demonstrate its robustness and efficiency in the specified applications. Future work is also identified.

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Copyright © 2008 by American Society of Mechanical Engineers
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Figures

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Figure 1

Flowchart to illustrate the proposed prototype-based initialization and growing training technique

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Figure 2

Ten types of nonlinear functions commonly seen in engineering mechanics applications and the recommended multilayer feedforward neural network architectures (i.e., prototypes) used to train them. Note that the indicated relationships are not exhaustive.

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Figure 3

A step-by-step procedure to illustrate the construction of the proposed (a) Prototype 1, variant a, (b) Prototype 2, variant a, and (c) Prototype 3, variant a. Three variants from each prototype are shown to the right of the procedure.

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Figure 4

Decomposing (a) swept sine and (b) multislope into a summation of some components that can be approximated directly using the proposed prototypes

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Figure 5

An example of combining Prototypes 1 and 2 to train a multislope function. The idea of decomposition is presented in Fig. 4. The target function is in gray, while those black curves with different line thicknesses show four random options using the Nguyen–Widrow initialization (6). Note that both Steps 1 and 2 were used to generate possible options for the initialization.

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Figure 6

Training results of a softening Duffing nonlinearity (30) based on two options using the Nguyen–Widrow algorithm and two other options using the proposed initialization methodology. All four trainings use neural network of five hidden nodes. The target function is in gray.

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Figure 7

Prototype 2, variant a (with four hidden nodes), is used to approximate various nonlinear functions. Note that some inputs and outputs are normalized. Also note that some of the trainings stopped prematurely.

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Figure 8

An example of using Prototype 2, variant a (with four hidden nodes), to approximate an idealized piecewise unsymmetrical nonlinearity with an offset that is typical of concrete in compression. The target function is in gray, while those black curves with different line thicknesses show four random options using the Nguyen–Widrow initialization (6). Note that both Steps 2 and 1 were gone through individually to generate three possible options for the initialization.

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Figure 9

An example of training performance as affected by the selection of the IW values. Here the target function (shown in gray) is a clearance (dead space) nonlinear function, while the three variants of Prototype 2 (with four hidden nodes) with and without proportioning during Step 1 of Stage III are selected for training. Note that some of the trainings stopped prematurely.

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Figure 10

An example of training performance as affected by the selection of the LW values. Note that the input is normalized while the output is not. The target functions are in gray, while those in black with different line types show various options created by scaling up the LW values of Prototype 2, variant a (with four hidden nodes). Note that some of the trainings stopped prematurely. Also note that normalized mean square error (NMSE) is used in the last panel for convenience of comparison.

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