Interaction-Aware Sensitivity Analysis for Aerodynamic Optimization Results using Information Theory

TL;DR

The power of novel information-theoretic approaches in identifying relevant parameters in optimization runs is demonstrated and how these methods avoid the selection of redundant parameters, while detecting interactions that result in synergistic contributions of multiple parameters is highlighted.

Abstract

An important issue during an engineering design process is to develop an understanding which design parameters have the most influence on the performance. Especially in the context of optimization approaches this knowledge is crucial in order to realize an efficient design process and achieve high-performing results. Information theory provides powerful tools to investigate these relationships because measures are model-free and thus also capture non-linear relationships, while requiring only minimal assumptions on the input data. We therefore propose to use recently introduced information-theoretic methods and estimation algorithms to find the most influential input parameters in optimization results. The proposed methods are in particular able to account for interactions between parameters, which are often neglected but may lead to redundant or synergistic contributions of multiple parameters. We demonstrate the application of these methods on optimization data from aerospace engineering, where we first identify the most relevant optimization parameters using a recently introduced information-theoretic feature-selection algorithm that accounts for interactions between parameters. Second, we use the novel partial information decomposition (PID) framework that allows to quantify redundant and synergistic contributions between selected parameters with respect to the optimization outcome to identify parameter interactions. We thus demonstrate the power of novel information-theoretic approaches in identifying relevant parameters in optimization runs and highlight how these methods avoid the selection of redundant parameters, while detecting interactions that result in synergistic contributions of multiple parameters.

Figures (5)

• Figure 1: Turbofan blade, modification and feature sets. A Investigated turbofan blade and Honda HF 120 jet engine, orange lines indicate cross-sections at which shape modifications were performed (LE: leading edge, TE: trailing edge). B Features selected form one blade section. C Modification of one blade section through addition of Hicks-Henne functions used during the shape optimization. D Location of the extracted feature sets, defined by varying numbers of sectional cuts through the geometry and number of points per section (red markers). Blue markers indicate leading and trailing edge features, each comprising two features for the $x$- and $y$-coordinate of the edge, respectively.
• Figure 2: Raw fitness values over generations for four investigated optimizations.
• Figure 3: A Partial information decomposition diagram: decomposition of the joint mutual information, $I(Y;X,Z)$ into unique information of each input variable (light and dark blue), redundant information (green), and synergistic information (red). B Corresponding, classical information-theoretic terms.
• Figure 4: Locations of selected features and identified interactions for four runs (rows A to D). Each column shows a different feature set, using 18, 21, 45, or 120 features respectively. Colored markers indicate selected features, dotted lines indicate the three pairs with the highest interaction wrt. the blade's fitness as measured by the synergistic information. The meaning of the colors is the same as in Fig. \ref{['fig:turbofan_parametrization']}D.
• Figure 5: Validation of feature set through $k$-nearest-neighbor-prediction of optimization outcome from selected feature sets (mean absolute error, MEA, y-axis). The x-axis indicates the size of the feature set, $|\mathbf{S}|$ (see main text). Each row shows results for one optimization run A-D. Each column shows one feature set with $N_{feat}=18$, $21$, $45$, and $120$ features. Orange markers indicate prediction results for FEAST feature sets of different size, blue markers indicate prediction results from machine learning approaches. Blue crosses are the feature sets selected with LARS while red crosses are those selected by MRMR. The black star ($\bigstar$) indicates results from our algorithm. Annotations indicate the two cases where another method gave better prediction results than our method for a smaller or equally large feature set.