## 1 Introduction  
Data-driven methods are increasingly being used in a wide variety of scientific fields. These methods extract knowledge and insights from datasets, which are typically large, and use this acquired knowledge to forecast new outputs. Artificial neural network (ANN) modeling is one of these data-driven methods that resembles the work principle of the human brain. An ANN acquires knowledge through a learning process and stores this knowledge through interneuron connections of different synaptic weight [1]. The ANN learns how a system behaves based on experience gained from the input/output dataset (the training process), and based on that, it can predict the system's output(s) for any new set of inputs. The ability of the ANNs to learn by example, makes them particularly useful for modeling highly complicated and non-linear processes, since the development of analytical models for such processes is extremely difficult. ANN modeling has been successfully used in a wide variety of engineering applications including automatic control [2], solar energy systems [3; 4], traffic and transportation [5], image processing [6], biomechanics [7], materials science and engineering [8; 9; 10], manufacturing [11; 12], fracture mechanics and fault detection [13; 14; 15; 16; 17; 18; 19; 20; 21; 22; 23; 24; 25; 26; 27]. A typical ANN consists of an input layer, an output layer and one or more hidden layers in between. Networks with up to two hidden layers are typically referred to as shallow neural networks (SNNs). They are the simplest type of networks and the most widely used in engineering applications. Networks with three or more hidden layers are referred to as deep neural networks (DNNs). They generally have higher prediction accuracy, and nowadays, they are attracting the most interest in the field of machine learning [14]. However, DNNs typically require large datasets for training the network in order to achieve high prediction accuracy for the unseen datasets. For applications where the number of inputs is relatively large, datasets comprising \(>\)10\({}^{4}\) data points are typically required [14]. Collecting and assembling large datasets is challenging, or even not feasible, in many applications, such as fracture mechanics and material science. For this reason, DNNs are still not finding a lot of use in some engineering applications. Most SNNs utilize only one hidden layer, and they generally do not require training datasets as large as those needed for DNNs. But nevertheless, even with a single hidden layer network, datasets comprising hundreds of data points are typically required to successfully train the network in order to achieve high prediction accuracy for the unseen dataset. It can be generally stated that the more complex and non-linear the relationships being modeled by the ANN are, the larger the size of the dataset required for training the network is. In fracture mechanics, ANNs were mostly used in applications concerned with crack propagation, fatigue life, and failure mode prediction [13]. In the field of mechanical fracture, ANNs have not found as much use as in other fields, mainly because it is not easy to generate large experimental datasets for training the ANN. This can mainly be attributed to practical constraints related to the time and cost requirements in many mechanical fracture experiments. Multiple site damage (MSD) cracks are small fatigue cracks that may accumulate at the sides of highly loaded rivet holes in the fuselage of an airplane. These MSD cracks usually appear after an extended period of time due to the large number of loading cycles, and therefore they present a major concern for aging aircraft fleets. Aircraft manufacturers design the fuselage of their airplanes to be able to carry the design load with the presence of a relatively large crack spanning several adjacent rivet holes. However, the presence of MSD cracks can significantly reduce the structure's ability to carry loads [28].