Confidence Intervals for Neural Networks and Applications to Modeling Engineering Materials

Abstract

Feedforward neural networks have been theoretically proved to be able to approximate a nonlinear function to any degree of accuracy as long as enough nodes exist in the hidden layer(s) (Hornik et. al. 1989). However, when feedforward neural networks are applied to modeling physical systems in the real world, people care more about their prediction capabilities than accurate modeling abilities. If a neural network is trained with noisy data measured from an experiment, what is the predictive performance of the neural network when unseen input data is fed into it? In this chapter, the confidence interval and prediction interval of a neural network model will be discussed. In particular, how the nonlinear structure of a feedforward neural network, impacts the confidence interval will be analyzed. Then, as an application, the measure of confidence to estimate nonlinear elastic behavior of reinforced soil is demonstrated. This chapter starts with a description of the structure of feedforward neural networks and basic learning algorithms. Then, nonlinear regression and its implementation within the nonlinear structure like a feedforward neural network will be discussed. The presentation will show confidence intervals and prediction intervals as well as applying them to a onehidden-layer feedforward neural network with one, two or more hidden node(s). Next, it is proceeded to apply the concepts of confidence intervals to solving a practical problem, prediction of the constitutive parameters of reinforced soil that is considered as composite material mixed with soil, geofiber and lime powder. Prediction intervals for the practical case is examined so that more quality information on the performance of reinforced soil for better decision-making and continuous improvement of construction material designs can be provided. Finally, the neural network-based parameter sensitivities will be analyzed. In order to clearly present the algorithms discussed in this chapter, some notations are declared as follows: matrices and vectors are written in boldface letters, and scalars in italics. Vectors are defined in column vectors. The superscript T of a matrix (or vector) denotes the transpose of the matrix (or vector).