Redes neurais artifciais: uma abordagem didática e aplicações em problemas de matemática aplicada

Abstract

This work presents an introductory study on arti cial neural networks, with emphasis on the mathematical foundations that support their structure and learning process. The main objective was to understand how these models operate from a mathematical perspective, especially highlighting the role of linear combination and gradient in the construction and training of neural networks. To this end, concepts from linear algebra, di erential calculus, and analytic geometry were revisited, establishing an appropriate theoretical basis for the interpretation of the arti cial neuron, activation functions, the cost function, and the backpropagation algorithm. Initially, the modeling of the arti cial neuron was discussed based on the linear combination of inputs, weights, and bias, emphasizing its relationship with lines, hyperplanes, and decision boundaries. Next, some of the main activation functions used in neural networks were presented, highlighting the importance of nonlinearity for the representational capacity of the model. Subsequently, the cost function, gradient, and chain rule were addressed, showing how these elements are articulated in the backpropagation process and in the updating of parameters through the gradient descent method. As a way of concretely illustrating the concepts studied, two computational applications were developed in Python. The rst consisted of a linear tting problem involving temperatures in degrees Celsius and Fahrenheit, in which the network successfully learned the expected relationship between input and output. The second involved a nonlinear classi cation problem in the plane, in which the network had to distinguish points located inside and outside a circle. In this case, the importance of the hidden layer and nonlinear activation functions for learning a decision boundary more compatible with the geometry of the problem was observed. In both applications, the matrix formulation proved particularly useful for organizing the calculations and supporting the computational implementation. Thus, it is concluded that the study of neural networks may constitute a relevant opportunity to connect Mathematics with contemporary topics in computing, contributing to a more grounded understanding of these models and opening possibilities for future developments in more general architectures and new applications.