Functional estimation problem mathematical description

In this study the problem of estimating an unknown function from its examples is revisited. Its mathematical description is attempted to map as closely as possible the practical problem that the potential NN user has to face. The objective of the chapter is twofold (1) to draw the framework in which NN solutions to the problem can be developed and studied, and (2) to show how careful considerations on the fundamental issues naturally lead to the Wave-Net solution. The analysis will not only attempt to justify the development of the Wave-Net, but will also refine its operational characteristics. The motivation for studying the functional estimation problem is the derivation of a modeling framework suitable for process control. The applicability of the derived solution, however, is not limited to control implementations. 

Identification of the material properties as an estimation of transfer function (TF) for the black box model. In this case the problem of identification is solving according to the results of the input (IN) and output (OUT) actions. There is a transfer of notion of mathematical description of TF on characterization of the material. This logical substitution gives us an opportunity to formalize testing procedure and describe the material as a set of formulae, which can be used for quantitative and qualitative characterization of the materials. 

Part I comprises three chapters that motivate the study of optimization by giving examples of different types of problems that may be encountered in chemical engineering. After discussing the three components in the previous list, we describe six steps that must be used in solving an optimization problem. A potential user of optimization must be able to translate a verbal description of the problem into the appropriate mathematical description. He or she should also understand how the problem formulation influences its solvability. We show how problem simplification, sensitivity analysis, and estimating the unknown parameters in models are important steps in model building. Chapter 3 discusses how the objective function should be developed. We focus on economic factors in this chapter and present several alternative methods of evaluating profitability. 

Density functional theory (DFT) is an enticing subject It appeals to chemists and physicists alike, and it is entrancing for those who like to work on mathematical physical aspects of problems, for those who relish computing observable properties from theory, and for those who most enjoy developing correct qualitative descriptions of phenomena. It is this combination of a qualitative model that at the same time furnishes quantitative reliable estimates that makes DFT particularly attractive for chemists. 

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