Abstract

In this work a conceptual theory of neural networks (NNs) from the perspective of functional analysis and variational calculus is presented. Within this formulation, the learning problem for the multilayer perceptron lies in terms of finding a function, which is an extremal for some functional. Therefore, a variational formulation for NNs provides a direct method for the solution of variational problems.

This proposed method is then applied to distinct types of engineering problems. In particular a shape design, an optimal control and an inverse problem are considered. The selected examples can be solved analytically, which enables a fair comparison with the NN results. Copyright © 2008 John Wiley & Sons, Ltd.

Abstract

Abstract

In this work an extended class of multilayer perceptron is presented. This includes independent parameters, boundary conditions and lower and upper bounds. In some cases, such extensions contain a priori information of the problem. On some other situations they are necessary in order to define a correct representation for the solution.

The use of this augmented class of neural network is illustrated through a case study in the optimal control theory. The numerical results are compared against the analytical solution.

Abstract

In this work we present a theory of the multilayer perceptron from the perspective of functional analysis and variational calculus. Within this formulation, the learning problem for the multilayer perceptron lies in terms of finding a function which is an extremal for some functional. As we will see, a variational formulation for the multilayer perceptron provides a direct method for the solution of general variational problems, in any dimension and up to any degree of accuracy. In order to validate this technique we use a multilayer perceptron to solve some classical problems in the calculus of variations.

Abstract

This research aims at analyzing bank credit of legal entity (in non-default, default and temporarily default), for the purpose of assisting the decision made by the analyst of this area. For that, we used Artificial Neural Networks (ANNs), more specifically, the Multilayer Perceptron (MLP) and the Radial Basis Functions (RBF) and, also, the statistical model of Logistic Regression (LR). For the implementation of the ANNs and LR, the softwares MATLAB and SPSS were used, respectively. For the simulations developed 5.432 data with 15 attributes were collected by the experts of the institution bank (called “XYZ”). The results show that the default clients are easily identifiable, but for the nondelinquent clients and for the temporarily defaulters, the techniques had greater difficulty in the discrimination, suggesting that they are no so discriminants. The main contributions of this work are: the analysis of three classes of clients (non-default, default and temporarily default), rather than just two (non-default and default) as is usually done; the coding of variables (attributes) of the company XYZ aiming to maximize the accuracy of the techniques and the use of the one-against all method, little used by the researchers of this research area. This work presents new insights towards research over Credit Risk Assessment showing other possibilities of client classification and codification, allowing different types of studies to take place.

Abstract

During each aircraft program a vast amount of aerodynamics data has to be generated to judge performance, structural loads as well as handling qualities. Within the past years the usage of computational fluid dynamics has significantly increased providing accurate insights into aircraft behaviour at early design stages and therefore at least partially enabled the mitigation of costly design changes. However, fully relying on high fidelity aerodynamic data is still computational prohibitive. Hence, data-driven models have gained an increasing attention in recent years. These methods not only provide continuous models but also enable the inclusion of highly accurate aerodynamic results in time-critical environments. This paper aims at applying deep learning techniques to derive such models and compare them to state of the art reduced order modeling techniques. In particular, three deep learning methods, a Multilayer perceptron for distribution predictions, a Multi-layer perceptron for pointwise predictions and an Autoencoder coupled with an interpolation technique are compared to Proper Orthogonal Decomposition and Isomap with latent space interpolation. For all methods an efficient methodology to determine hyperparameters is outlined and applied. Results are presented for an Airbus provided XRF1 dataset which includes surface pressure distributions at various Mach numbers and angles of attack.