Black box modeling

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. 

An extreme case of these empirical models are black box models, predominantly polynomials, the application of which is strictly restricted to the range of operating conditions and design variables for which the models were developed. Even in this range, optimization using black box models can lead to operating conditions far from the real optimum. This is due to non-linearities of the real systems, which cannot be modelled by polynomials. Black box... 

Those based on strictly empirical descriptions (so-called black box models). 

The electrostatic precipitator in Example 2.2 is typical of industrial processes the operation of most process equipment is so complicated that application of fundamental physical laws may not produce a suitable model. For example, thermodynamic or chemical kinetics data may be required in such a model but may not be available. On the other hand, although the development of black box models may require less effort and the resulting models may be simpler in form, empirical models are usually only relevant for restricted ranges of operation and scale-up. Thus, a model such as ESP model 1 might need to be completely reformulated for a different size range of particulate matter or for a different type of coal. You might have to use a series of black box models to achieve suitable accuracy for different operating conditions. 

A convenient approach to the case is the use of ANNs, as first demonstrated by the seminal work by Bos and Van der Linden [46]. ANNs generate black-box models, which have shown special abilities to describe nonlinear responses obtained with sensors of different families. Unfortunately, these tools create models only from a large amount of departure information, the training set, which must be carefully obtained [47]. The extra information is in account of the absence of a thermodynamical or physical model, for example the Nicolsky-Eisenman equation. 

Bove R., Lunghi P., Sammes N.M., 2005. SOFC mathematic model for systems simulations. Part One From a micro-detailed to macro-black-box model. International Journal of Hydrogen Energy, 30(2), 181-187. 

Figure 2.2 shows the black-box model. The inlets indicated by arrows X1 X2,..., X]< are the possibilities of affecting the research subject. The outlet arrows yi, y2,-., ym or outlets are responses, optimization criteria or aim Junctions. 

Gray-box modeling. ANNs can learn complex functional relations by generalizing from a limited amount of training data. When part of the process is well understood, white- and black-box models can be combined and called gray-box modeling. 

Physical state space models are more attractive for use with the LQP (especially when state variables are directly measurable), while multivariable black box models are probably better treated by frequency response methods (22) or minimum variance control (discussed later in this section). 

In practice, one is often faced with choosing a model that is easily interpretable but may not approximate a response very well, such as a low-order polynomial regression, or with choosing a black box model, such as the random-function model in equations (l)-(3). Our approach makes this blackbox model interpretable in two ways (a) the ANOVA decomposition provides a quantitative screening of the low-order effects, and (b) the important effects can be visualized. By comparison, in a low-order polynomial regression model, the relationship between input variables and an output variable is more direct. Unfortunately, as we have seen, the complexities of a computer code may be too subtle for such simple approximating models. 

At present, however, most TK studies in ecotoxicology focus on uptake and elimination kinetics and can be modeled only by the DBTK approach. These studies generally only address the uptake and elimination of the chemicals that are detectable and how these are influenced by the other chemicals and still represent black box models. However, for many applications, this may be sufficient. 

Derivation The statistical modeling method must be correctly applied. Black box models, such as artificial neural networks, are less suitable for clinical applications... 

There is a growing interest in designing and optimizing process and products using complex mathematical models which has produced a large number of specialized software. However, the arquitecture of most of these programs is modular in order to use tailored numerical methods developed or adapted to each particular problem. In most situations, the final user only can view a black box model with limited access to the original code. 

Using optimization algorithms with these black box models is a challenging problem for at least two reasons. First, some of those models can take significant CPU computation and second, derivatives for gradient based algorithms usually cannot be accurately estimated because most of these black box models introduce noise (small sensitivity of some variables, termination criteria in the algorithms, etc). 

In this work, we develop an algorithm based on fitting response surfaces -using a kriging metamodel- for the optimization of constrained-noise black box models. Besides, an important characteristic is that we deal with constrained problems in which the metamodel can represent either the objective function or some constraints (or both simultaneously). A typical case is the optimization of process flowsheets using modular simulators in which some units are represented by a metamodel. In these systems it is possible to include external constraints and even the result of some calculations could be constraints to the model. 

Although the algorithm described in next paragraphs is described for optimizing process flowsheets, it can be used for any black box model with or without noise. It guarantees a local minimum within a pre-specified tolerance. 

---