# A Multifeatured Data-Driven Homogenization for Heterogeneous Elastic Solids
## 1 Introduction
[19] developed an autoregressive model to capture the load-deflection behavior of materials. The recurrent neural network was first utilized for training the mechanical behavior of materials [20], in which the water pressure associated with strain and porosity was adopted as inputs. [PERSON] and [PERSON] [21] considered material properties as features for training the neural network, and then they predicted a load-deflection behavior. Similarly, [PERSON] and [PERSON] [22] used the neural network to learn constitutive laws; they used \(\sigma_{xx}\) and \(\sigma_{zz}\) aligned with porosity and friction angle for the prediction of soil behavior. AI methods have also been used for homogenization in the multiscale aspects to reduce computational calculations. In 2001, [PERSON] et al. [23] developed a pre-trained material model based on Artificial Neural Network (ANN) to capture the nonlinear and damage behavior of heterogeneous materials. This approach utilized the strain, geometry, and damage information as input and the stress as output for training the micromechanical behavior. Recently, [PERSON] et al. [24] trained path-dependent behavior of a material using a stacked long-short term memory unit (LSTM) for the first time. They predicted the total stress tensor provided the full strain tensor, which is beneficial for concurrent multiscale methods. [PERSON] et al. [25] applied a machine learning method for plastic polycrystals by introducing descriptors that represent the geometry of microstructures. In contrast, unsupervised machine learning was also used for path-dependent behaviors. For example, [PERSON] et al. [26] developed a cooperative game for the automated learning of elastoplastic material responses. Regarding FE\({}^{2}\) scheme, [PERSON] and [PERSON] [27] developed a data-driven model for path-dependent materials using the LSTM, where the consistent tangent was calculated using the auto-differentiation of TensorFlow. Similarly, [PERSON] and [PERSON] [28] leveraged surrogate modeling to reduce the computational cost by establishing a direct relationship between the inputs and outputs of finite elements. In this study, a computational data-driven homogenization for heterogeneous materials is developed to efficiently incorporate the microscopic features in material and geometrical aspects. Here elastic responses is only considered (linear and nonlinear), and the single-phase system of porous solids is considered to focus on mechanical responses. A novel numerical experiment is designed to generate a database for micromechanical responses of the heterogeneous solids. The first design of experiment (DOE) includes various microstructures by randomly generating void spaces with different numbers and sizes. The geometrical features of microstructures are coined as descriptors, which are expressed by the probabilistic information of the pore structures. Then, different material properties are considered to account for the material heterogeneity. Diverse micromechanical behaviors are recorded via the homogenization scheme for FE\({}^{2}\), where the information of mechanical responses, including the strain and both material and geometrical information of microstructures, are considered inputs of the training process. Then, a Deep Neural Network system is established to predict the stress responses finally. The proposed model demonstrates an excellent capability of the homogenization method in capturing the elastic (linear and nonlinear) responses of heterogeneous porous solids. The organization of this paper is as follows. In Section 2, the double scale finite element method and the Deep Learning method are described to explain the theoretical background of the paper. In Section 3, the design of the experiment for generating data is presented in three aspects: (1) generation of heterogeneous microstructure, (2) extracting microstructural descriptors, and (3) generation of loading path.