## 3 Artificial Neural Network (ANN) Model for Structural Characterization  
ML is used in the proposed methodology of this paper as a means of computational statistics for prediction using computers. ML algorithms use computational statistics to find patterns in data such as numbers, words, images, etc. The ML model is a mathematical model that takes available data for training and performs prediction on its own based on supervised, unsupervised or reinforcement learning. NN is a special type of ML that is inspired by the human neuron structure. See an example in Figure 7 for NN. The NN models are computing systems considered similar to biological neural networks that form human and animal brains. The NN is considered as a function with an input and a desired output, where an output is dependent on weighted inputs. The following are some of the types of validations using the ML model for the certification of coupled multifunctional properties of AI-enabled and additively manufactured multifunctional carbon fiber composites. The proposed new method converts the detailed experimental design, test plan, test matrix and other data, such as failure loads, failure modes, failure locations, material properties, processes, etc., into a simplified version of the computational model as shown in the below Table 1 as inputs, NN model and outputs. The method uses an ANN model for characterizing the strength of AI-enabled and additively manufactured multifunctional and continuous carbon fiber composite structures. Table 1 shows the list of inputs, ANN models, and the associated outputs. The use of ANN for a particular set of inputs determines the outputs. The anisotropic behavior and the defect vulnerabilities of the composite material systems due to variabilities in the material characterization properties make the composites more process dependent. The use of ANN in the design, development, additive manufacturing and certification of multifunctional composites for aircraft, drones and spacecraft appears to be the most desired application of ANN to achieve the safety, integrity and life extension of the aerospace vehicles, since the composite material systems are highly process-dependent. The use of ANN in the design, development, manufacturing, testing and certification of multifunctional composites for aerospace applications can be leveraged to minimize the presence of flaws and defects in the advanced composites that are inherently present due to environmental, material and processing factors.  
Figure 7: Structure of an artificial neural network (ANN) with 3 hidden layers.  
Mathematically, as shown in Equation (1), the ANN function \((Y)\) is known as a composition of other functions, \(f(Q_{j})\), which can be further decomposed into other functions. The ANN function \((Y)\), represented as a neural network structure with arrows depicting the relationships and dependencies between different functions, is shown below in Figure 7. The ANN computation is conducted within a fraction of a section depending on the computing architectures of the system and the number of hidden layers for the ANN. A typical ANN model includes a minimum of three layers: input, hidden and output. Each neuron in the ANN model performs a weighted summation of the inputs, which then passes a nonlinear activation function \(Q_{j}\), also called a neuron function, and finally, the data are fed out of the neural network through an output layer [37]. Figure 7 shows only three hidden layers for ANN computation. The use of the higher number of hidden ANN layers will yield a higher degree and complexity of the mathematical calculations, thereby resulting in robust outputs and the need for higher computing infrastructures.