Pareto Optimization of GMDH-Type Neural Networks for Modelling and Prediction of Hoop Strain in Explosive Forming Process

Abstract

In this paper, Evolutionary Algorithms (EAs) are deployed for multi-objective Pareto optimal design of Group Method of Data Handling (GMDH)-type neural networks that have been used for modelling of the effect of dynamic yield stress, thickness, charge mass, centre deflection and distance from centre on the hoop strain in explosive forming process using some input-output experimental data. In this way, EAs with a new encoding scheme is firstly presented to evolutionary design of the generalized GMDH-type neural networks in which the connectivity configurations in such networks are not limited to adjacent layers. Multi-objective EAs (non–dominated sorting genetic algorithm, NSGA-II) with a new diversity preserving mechanism are secondly used for Pareto optimization of such GMDH-type neural networks. The important conflicting objectives of GMDH-type neural networks that are considered in this work are, namely, Training Error (TE), Prediction Error (PE) and number of neurons (N) of such neural network. Different pairs of these objective functions are selected for 2-objective optimization processes. Therefore, optimal Pareto fronts of such models are obtained in each case which exhibit the trade-offs between the corresponding pair of conflicting objectives and, thus, provide different non-dominated optimal choices of GMDH-type neural networks models for the explosive forming process. Moreover, all the three objectives are also considered in a 3-objective optimization process which consequently lead to some more non-dominated choices of GMDH-type models representing the trade-offs among the training error, prediction error, and number of neurons (complexity of network), simultaneously. The overlay graphs of these Pareto fronts also reveal that the 3-objective results include those of the 2-objective results and, thus, provide more optimal choices for the multi-objective design of GMDH-type neural networks in terms of minimum training error, minimum prediction error and minimum complexity.