Mathematical models, learning purpose

Numeric-to-numeric transformations are used as empirical mathematical models where the adaptive characteristics of neural networks learn to map between numeric sets of input-output data. In these modehng apphcations, neural networks are used as an alternative to traditional data regression schemes based on regression of plant data. Backpropagation networks have been widely used for this purpose. 

The purpose of the Section 3.1 is to analyze the implications for mathematical models, each of a practical value, because of the introduction of this force term. A number of known fluid mechanics models can be generalized in such a way. Their consideration highlights most of the properties of the obstructed flows, allows the verification of this theoretical approach, and is a useful methodological tool for teaching and learning the general problem. 

A fundamentally different type of simulation is offered by science-based learning environments. Such environments incorporate some general-purpose mathematical engine that either represents nature directly or that can be programmed to represent nature. Examples are Mathematica (7) and some similar programs (8-9) for general analytical modeling in the physical sciences and Interactive Physics for introductory classical mechanics (10). Mathematica and Interactive Physics can be applied to countless topics, as opposed to the narrow focus of Flash-based simulations. Even more importantly, Mathematica and Interactive Physics are open-ended in that the software may accommodate unscripted inquiries and follow-up questions. 

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