[pt] Este trabalho apresenta uma abordagem inovadora para calcular autopares de operadores diferenciais (OD), utilizando programação genética (PG) e computação simbólica. Na literatura atual, o Método dos Elementos Finitos (MEF) e o Método das Diferenças Finitas (MDF) são os mais utilizados. Tais métodos usam discretização para converter o OD em uma matriz finita e, por isso, apresentam limitações como perda de acurácia e presença de soluções espúrias. Além disso, se o domínio do OD fosse alterado, os autopares precisariam ser calculados novamente, pois a representação matricial do operador depende dos parâmetros do problema. Nesse contexto, este trabalho propõe evoluir autofunções analiticamente usando PG, sem discretização do domínio. Com isso, é possível incorporar parâmetros, o que torna a solução obtida válida para uma classe de problemas. Este texto descreve o modelo para OD normais, aplicando conceitos de indivíduos multi-árvore e diferenciação simbólica. O modelo evolui auto-funções e, a partir delas, calcula os autovalores empregando a razão de Rayleigh. Experimentos baseados em aplicações da Física mostram que a técnica proposta é capaz de encontrar as autofunções analíticas com a acurácia igual ou melhor que as técnicas numéricas supracitadas. Finalmente, a técnica proposta é aplicada ao problema de propagação de ondas eletromagnéticas em poços de petróleo em ULF e UHF. As soluções analíticas são dadas em função do diâmetro e do percentual de incrustações no poço. Os resultados mostram que, para um conjunto suficientemente grande de valores distintos dos parâmetros, a técnica apresenta tempo de execução inferior às técnicas clássicas, mantendo a acurácia destas. / [en] This work presents an innovative approach to calculate the eigenpairs of linear differential operators (LDO), employing genetic programming (GP) and symbolic computation. In the current literature, the Finite Element Method (FEM) and the Finite Difference Method (FDM) are more commonly used. Such methods use discretization to convert the LDO to a finite matrix, therefore causing loss of accuracy and presence of spurious solutions. Additionally, if the domain of the LDO was changed, the eigenpairs would need to be recalculated, since the matrix representation of the LDO depends on the parameters of the problem. In this context, this work proposes to evolve eigenfunctions analytically using GP, without domain discretization. Hence, it is possible to incorporate the parameter, which makes a obtained solution valid for a class of problems. This text describes the model for normal LDO, applying concepts of multi-tree individuals and symbolic differentiation. The presented model evolves eigenfunctions and, then, calculates the eigenvalues using the Rayleigh quotient. Experiments based on Physics problems show that the proposed technique is able to find the analytical eigenfunctions with the same accuracy of the numerical techniques mentioned above. Finally, the proposed technique is applied to the problem of propagation of electromagnetic waves in oil wells in ULF and UHF. The analytical solutions are given as a function of the diameter and percentage of CaCO in the well. The results show that, for a sufficiently large set of distinct values of the parameters, the technique presents execution time inferior to the FEM, while maintaining its accuracy.

The dissertation thesis investigates the modeling of the neural network activity with a focus on a multilayer forward neural network (MLP – Multi Layer Perceptron). In this often used structure of neural networks, time-varying neurons are used, along with an analogy in modeling hyperstructures of linear differential operators. Using a finite lemma and defined hyperoperation, a hyperstructure composed of neurons is defined for a given transient function. There are examined their properties with an emphasis on structures with a layout.

In the thesis we discuss binary hyperstructures of linear differential operators of the second order both in general and (inspired by models of specific time processes) in a special case of the Jacobi form. We also study binary hyperstructures constructed from distributive lattices and suggest transfer of this construction to n-ary hyperstructures. We use these hyperstructures to construct multiautomata and quasi-multiautomata. The input sets of all these automata structures are constructed so that the transfer of information for certain specific modeling time functions is facilitated. For this reason we use smooth positive functions or vectors components of which are real numbers or smooth positive functions. The above hyperstructures are state-sets of these automata structures. Finally, we investigate various types of compositions of the above multiautomata and quasi-multiautomata. In order to this we have to generalize the classical definitions of Dörfler. While some of the concepts can be transferred to the hyperstructure context rather easily, in the case of Cartesian composition the attempt to generalize it leads to some interesting results.