Evolução de modelos diferenciais para sistemas concretos por Programação Genética

Abstract

A system is defined by its entities and their interrelations in an environment which is determined by an arbitrary boundary. Complex systems exhibit emer- gent behaviour without a central controller. Concrete systems designate the ones observable in reality. A model allows us to understand, to control and to predict behaviour of the system. A differential model from a system could be understood as some sort of underlying physical law depicted by either one or a set of differential equations. This work aims to investigate and implement methods to perform computer-automated system modelling. This thesis could be divided into three main stages: (1) developments of a computer-automated numerical solver for linear differential equations, partial or ordinary, based on the matrix formulation for an own customization of the Ritz-Galerkin method; (2) proposition of a fitness evaluation scheme which benefits from the devel- oped numerical solver to guide evolution of differential models for concrete complex systems; (3) preliminary implementations of a genetic programming application to perform computer-automated system modelling. In the first stage, it is shown how the proposed solver uses Jacobi orthogonal polynomials as a complete basis for the Galerkin method and how the solver deals with auxiliary conditions of several types. Polynomial approximate solutions are achieved for several types of linear partial differential equations, including hy- perbolic, parabolic and elliptic problems. In the second stage, the proposed fitness evaluation scheme is developed to exploit some characteristics from the proposed solver and to perform piecewise polynomial approximations in or- der to evaluate differential individuals from a given evolutionary algorithm population. Finally, a preliminary implementation of a genetic programming application is presented and some issues are discussed to enable a better un- derstanding of computer-automated system modelling. Indications for some promising subjects for future continuation researches are also addressed here, as how to expand this work to some classes of non-linear partial differential equations.