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Sectional Sensitivity Measures with Artificial Neural Networks

For the design of engineering structures and the assessment of their reliability it is of main interest to appraise the importance of input quantities in view of result quantities. Sensitivity analysis provides a versatile tool to assess those importance. In the past, this was done by determining the gradients of the function of interest and deduce sensitivity statements by means of partial derivatives in a local manner. For most engineering applications those procedures are inappropriate, since it is limited to linear functions and ignores the spreading of the respective input quantities. Thus, enhanced sensitivity measures are elaborated, which assess the variances of functions in a global manner. Nevertheless, to further improve the informative value of sensitivity statements global partial derivative based sensitivity measures are introduced. However, due to the computational expense of sophisticated sensitivity analysis, efficient analysis methods are in steady advance. A main focus is thereby on the coupling of metamodeling techniques and sensitivity analysis. Generally, sensitivity measures condense available information of global input spaces to singleton values. In consequence, the importance of local parts of the input spaces are neglected and the characteristic of the functional relationship between input and result parameters remains hidden. While this fact can be neglected in high dimensional problems, it is of main interest in low dimensional problems. This becomes obvious, considering the various software tools providing metamodel viewer. Thereby, metamodel viewer suffer the dimensionality problem; while two parameters can be visualized, the re- maining parameters are fixed to discrete parameter values. Thus, modifying those parameters will alter the visualized response surface and disturb deduced information. Especially for challenging problems the evaluation of metamodel viewer results may become cumbersome. Nevertheless, for good-natured problems with few sensitive parameters those metamodel viewers are useful to derive an idea about the behavior of the function of interest. Alternatively, in this paper the approach of sectional sensitivity measures is introduced, which does not feature the dimensionality problem. Thereby, the global sensitivity analysis is extended to provide sensitivity information in specific parts of the input space. Merging all those information together, statements about functional dependencies are obtained. Thereby, sectional sensitivity measures can be distin- guished in argument based sectional sensitivity measures and sectional sensitivity measures based on the value of function. While the former is proper to deduce statements about the functional relationship between individual input and result parameters, which equals the idea of metamodel viewer, the latter is deployable for reasoning statements about influences of input parameters in specific regions of the result space. However, the generated sensitivity statements are not only useful for data mining purposes, they can be even utilized to advance optimization and reliability tasks. While sophisticated sensitivity approaches provide worthwhile results, their computational expense hinders sometimes the applicability for industry-relevant problems. Thus, sensitivity analysis may be coupled with metamodels. Here, artificial neural networks are applied. Neural networks are capable to reason unknown dependencies between variables on the basis of a set of initial information. Thereby, the information content is stored within the neural network. Utilizing the respective properties, statements about the sensitivity may be derived. Thereby, the sensitivity is assessed by means of measures capitalizing diverse properties of the artificial neural network. Those are data handling, derivability and efficient numerical evaluation. In result, multi-faceted sensitivity measures may be defined or evaluated like weighting-based, derivative-based and even variance-based measures. The appropriateness of the novel approaches is demonstrated by means of analytical functions and their applicability is shown by means of an industry-relevant example.

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