Connectionist Models for Solving Linear and Nonlinear Equations

Abstract

Various engineering and science problems may transform into linear and nonlinear equations, in general. In recent decades, Artificial Neural Network (ANN) has emerged as one of the prominent mechanism for solving linear and nonlinear equations. Although linear and nonlinear equations may be solved by different known analytical and numerical methods but those are sometimes having different complexity to handle. Traditional numerical methods may sometimes fail to solve these equations due to the involvement of singularities or complexity of the function etc. Moreover, (for example) there may exist two closely positioned roots or due to discontinuity of the curve in the problems of root finding and then the known numerical methods may sometimes difficult to use. In case of linear system of equations, the traditional numerical methods sometimes fail if the system is not diagonally dominant, positive definite etc. In those cases, ANN based methods may be an alternative for solving the equations. In this regard, detail ANN procedure with various example problems related to transcendental, Diophantine and linear system of equations with their network architectures have been addressed here to demonstrate the proposed procedure.
Further, solving linear and nonlinear eigenvalue problems are also challenging task. For example, dynamic analysis of structure without damping may transform into a linear eigenvalue problem and with damping it leads to a nonlinear eigenvalue problem. Linear eigenvalue problems are studied though by many authors, but nonlinear eigenvalue problems are not studied much. However, these methods (for both linear as well as nonlinear) may sometimes be problem dependent and difficult to handle. As such, in these cases, ANN may also be advantageous over the existing methods. Few examples of linear eigenvalue problems such as vibration analysis of spring mass system and multi-storey shear building have been investigated. On the other hand, two examples of overdamped spring mass systems have been examined to show the efficacy of the proposed method in case of nonlinear eigenvalue problem.
It may be noted that parameters involved in the above systems may not be crisp (exact) always because of errors in experiment, measurement and observation. In that case, the problem leads to an uncertain system. In order to handle these uncertainties, recently researchers have introduced interval and/or fuzzy numbers in place of crisp ones. In these regards, various techniques have been developed by different authors but these are sometimes valid for certain (particular) type of problems only. These methods may have few drawbacks that include number of iterations, triangularisation etc. In this context, application problems such as static problems of structures lead to system of equations. As mentioned earlier that inclusion of uncertainty makes the problem as uncertain. Similarly, computation of the interval controls using pole placement technique in case of uncertain plant system reduces to interval linear system, which itself is a challenging problem. Moreover, dynamic problems lead to eigenvalue problems which may become more complicated due to the inclusion of uncertainty. Accordingly, ANN methods have been developed to handle the above problems with ease. Different example problems have been solved in this context to validate the proposed ANN technique.