## 1 Introduction  
Artificial neural networks (ANNs) represent an approach for solving complex problems in numerous fields and are particularly useful for dealing with complex input-output relationships without explicit mathematical functions [1]. ANNs simulate the human brain to process information rapidly and efficiently through a system of neural networks consisting of vast numbers of neurons. ANNs usually consist of an input layer, an output layer and one or more hidden layers. In general, the number of neurons of the input layer must correspond to the number of input variables; the number of neurons of the output layer must be equivalent to the number of outputs produced by the network; and the hidden layer performs the operations designed to achieve the desired output, where its number of neurons is obtained by a process of trial and error [1,2]. The development of the ANN computingparadigm has been mainly driven by the practical requirement to solve specific problems in a rapid and more computationally-efficient manner. The ability to learn is a peculiar feature of intelligent systems. In artificial systems, learning is viewed as the process of updating the internal representation of a system in response to external stimuli so that it can perform a specific task with greater precision and efficiency. The learning process in ANNs includes modifying the network architecture, which involves incrementally adjusting the magnitudes of the weights of the connections between neurons in the network (also known as the \"synapse strength\") [2; 3]. This process is performed repetitively as the network is presented with training examples, which is similar to the process by which biological organisms learn from experience. As a result, ANNs have been one of the most powerful statistical approaches for solving problems that are not amenable to conventional statistical methods [3]. In contrast to the modelling of input-output relationships using ANN techniques, regression models provide a quantitative description of input-output relationships using specific mathematical functions. For the ultrasonic machining (USM) technique, process responses are associated with the selection of machining parameters [4; 5; 6; 7]. An increased material removal rate (MRR) has always been one of the primary goals of USM manufacturers and users [8; 9; 10; 11; 12]. A number of methods have been developed for selecting machining parameter values to enhance the MRR, such as Taguchi statistical analyses and regression modelling analyses [13; 14; 15; 16; 17; 18; 19]. In our past work [5], we analysed the MRR of USM in detail based on orthogonal tests using the Taguchi method. The regression modelling analyses employed three main machining parameters affecting the process response (i.e., abrasive granularity, feed pressure and feed speed). Analysis of variance (ANOVA) was executed to investigate the statistical significance of the parameters at a 95% confidence level and to determine the relative contribution of the parameters to the process response. The results showed that the abrasive granularity had a significant and nonlinear effect on the MRR in USM, where an abrasive granularity deviating substantially from an optimum value negatively affected the MRR. On this basis, an improved nonlinear regression model was established using a mathematical modelling method. The prediction precision of the improved nonlinear regression model was demonstrated to be substantially greater than that of a conventional linear regression model. However, identifying an appropriate function for fitting the nonlinear relationships between the machining parameters of USM and the process response is typically quite difficult. Therefore, this approach requires much more powerful modelling methods that can facilitate mapping between machining parameters and process responses. For this reason, ANNs have been applied for complex input-output modelling and have nearly always improved the correlations obtained relative to conventional regression methods [20; 21].