# Feedback-Forward Neural Networks for Failure Mechanics Problems
### Deep Learning (DL) Architectures  
DL has many architectures, especially in mechanics, and research is focused mainly on using the feed-forward neural network (FFNN), recurrent neural network (RNN), and convolutional neural network (CNN). So far, applications of these types of neural networks devoted to the field of mechanics can be seen in the recent works of [27; 28; 29; 30; 31; 32; 33; 34; 35; 36; 37; 38; 39; 40; 41; 42; 43]. A detailed description of those approaches is summarized as follows:  
* FFNN is used in [44; 45] to model the material behavior at the macroscale level, using strains (as inputs) and stresses (as outputs). One main advantage of such a model is that the training data required for the neural network can be directly acquired from experimental data. On this basis, and for modeling a neural network that approximates the non-linear behavior of history-dependent material models (e.g., plasticity, where loading history is relevant), Ref. [46] proposed the incorporation of the strain from the previous load step as an input data or feature for the neural network. Recently, a novel method called the proper orthogonal decomposition feed forward neural network (PODFNN) was proposed by [47] for predicting the stress sequences in the case of plasticity, which reduces the complexity of the model significantly by transforming the stress sequence into multiple independent coefficient sequences. * RNN is another type of neural network that uses the previous outputs as inputs, i.e., path-dependent scheme. Two different approaches, direct (black box) and graph-based (physically-informed), were applied in [37] for modeling elastoplastic materials. In the former case (black box approach), the total stress was predicted purely considering the total strain history, whereas in the latter case (graph-based approach), besides using the recurrent neural network to predict the path-dependent behavior, the feed forward neural network is used to predict the path-independent responses, which may lead to a more accurate prediction of stresses. * CNN is a special type of deep neural network that has recently become a dominant method in computer vision. A CNN architecture consists of an input and an output layer, as well as multiple hidden layers. These hidden layers typically consist of convolutional layers, activation layers, pooling layers, and fully connected layers. CNNs have been used in image classification, video classification, face recognition, scene labeling, action recognition, image segmentation, and natural language translation, among others. In the work of [48] CNNs are used to quantitatively predict the mechanical properties (i.e., stiffness, strength, and toughness) of a 2D checker-board composed of two different phases (brittle and ductile). Following this line, [PERSON]. [49] introduced a graph convolutional deep neural network, incorporating the non-Euclidean weighted graph data to predict the elastic response of materials with complex microstructures. For recent works on CNNs, we refer to [50; 51; 52], and the citations therein. Within this work, a feed-forward neural network (FFNN) is employed to perform the regression task of predicting stresses in the case of linear elastic material and also predicting the elasticity in the fracture phase-field modeling of brittle solids along with the prediction of the crack phase field. The objective of this paper is to incorporate neural network models as constitutive models within the finite element applications for elasticity and phase-field modeling of brittle fracture by combining the available tools. The efficiency of the trained neural network is studied within the finite element analysis. The raw numerical data acquired by FEM simulation are used to train the neural network model under investigation. As an advantage of this contribution, the proposed method has the potential for providing accurate and feasible approximations for various engineering applications.