# Efficient Deep Learning for Gradient-Enhanced Stress Dependent Damage Model
## 1 Introduction  
Neural networks (NN) have been used for numerous applications in different areas including computational mechanics. Initially, single-layer or shallow neural networks (SNN) consisted only of one input and one output layer. Later on, additional hidden layers were added to the network architecture resulting in so-called deep neural networks (DNN). Table 1 gives a brief summary of different network architectures. Neural networks or more specifically deep neural networks suffer from three major difficulties, i.e., (a) vanishing gradients, (b) over-fitting and (c) computational loading. However, significant advances such as deep belief networks (DBN) [1], rectified linear unit (ReLU) activation functions [2], drop-out algorithms [3] or back-propagation algorithms and associated tools have contributed to the popularity of DNN. The vanishing gradient problem for instance has been significantly alleviated thanks to the RELU activation function and cross entropy-driven learning techniques. Nonetheless, certain issues such as over-fitting still remains a challenge in deep neural networks. Common techniques to address such problems include regularization techniques. We recall that the crucial idea behind Artificial Neural Network (ANN) is that many neurons can be joined together by connecting weights to conduct complex computations. The structure is oftendemonstrated as a graph or a map whose nodes are the neurons and each (directed) edge in the graph connects the output to the input of the associated neurons. Deep learning methods, representative of learning methods with multiple processing layers in the hidden layers, consist of linear and non-linear transformations [4; 5]. Some simplifications of the problems stem from their need for a usually very strong CPU and a noticeably long time to detect and analyse slow convergence behaviour. For complex problems, the solution could be non-existent. A newer method or a faster algorithm has yet to be found in recent decades. While machine learning (ML) approaches have been extensively and successfully used in numerous areas, its application in \"modeling and simulation\" is still in its infancy. For instance, in medicine, ML has been employed in diagnostics where it outperforms the diagnosis of established physicians [6]. The authors of [7] have applied successfully deep learning to cellular imaging. [PERSON] et al. [8] focused on the problems of cellular imaging in regulatory genomics. [PERSON] et al. [9] took advantage of deep learning in computational chemistry. Deep learning techniques were also applied in applications such as bio-informatics, or the public health sector [10]. On the other hand, most approaches in engineering, or more specifically computational mechanics, have been used in data-driven contexts though there are numerous other applications such as the direct solution of partial differential equations [11], which have the potential to drastically accelerate the design to analysis time and the way modeling and simulation is performed. In the data-driven context, neural networks have commonly been used in the context of constitutive models [12; 13] as an alternative to traditional constitutive models. The models in [14; 15; 16] presented interesting approaches to capture the response of anisotropic materials. New training algorithms for specific constitutive laws have been presented in [17; 18]. However, setting up the network architecture for such engineering problems still remains a major challenge and is often determined by trial and error. The authors of [19] exploited Deep Learning to optimize the fine-scale structure of composites. Multi-fields problems were tackled for instance in [20; 21]. Recently, [PERSON] et al. [22] has applied deep learning algorithms to structural analysis. In this manuscript, we present a novel methodology to predict the load-deflection curve by deep learning. Passing through the three-point bending as an illustrative example, we suggest some possible architectures of the deep neural networks based on the Adam optimizer.