## 1 Introduction  
Multiscale modeling is a simulation technique that describes a behavior at a given length scale based on the physics at a finer scale, which in turn is considered to be better known or easy to model. Structures made of composite materials or materials with microstructure constitute typical candidates for multiscale modeling. In fact, it is impossible to model a whole structure by taking into consideration all constituents and their possible nonlinear interactions within the model. Therefore, a compromise must be considered wherein a reduction in precision and an increase in uncertainty related to multiscale modeling should be accepted in order to avoid the complexity of a full-scale model. In engineering, homogenization is a representative example of applied multiscale modeling [1]. In linear problems, homogenization is usually based on analytical expressions that give the homogenized properties for a given microstructure or composite material [2, 3]. This approach is powerful, despite certain theoretical difficulties for the calculation of homogenized properties. Numerical homogenization allows for the extension of classical analytical homogenization in order to describe more complicated problems. Homogenized properties are load dependent or even path dependent; therefore, analytical solutions are either difficult to calculate or do not exist. A path-independent problem is considered here as a first example. ANNs have been applied in different areas of science and engineering, where data sets are available for training and testing. In particular, ANNs are employed in elastoplastic and contact problems in mechanics by using a minimization of energy. The Hopfield and Tank neural networks have been proposed by [PERSON] and [PERSON] [4] and [PERSON] et al. [5]. The feedforward NNs trained by the backpropagation algorithm have been used for an approximation of several problems in mechanics based on examples (supervised learning). The buckling loads in nonlinear problems for elastic plates have been calculated by neural networks in [PERSON] and Stavroulakis [6]. Inverse and parameter identification problems in mechanics have been solved by using backpropagation neural networks in Stavroulakis [7]. A recent review of the classical usage of neural networks within computational mechanics has been published by [PERSON] and [PERSON] [8]. Several neural network-based constitutive models have been proposed in recent works. The main directions are described in the review article by [PERSON] et al. [9]. The work described here belongs to the more general topic of CANN (constitutive ANN). In fact, from input-output relations, an ANN is able to learn and replace a mathematically consistent constitutive material model. Adding additional information, like thermomechanical restrictions, enhances the effectiveness and allows for training with less data. Further information can be found in, among others, references [9; 10; 11]. Here, an efficient way to store the constitutive relation based on a database of responses from a representative volume element (RVE) is proposed through the construction of a neural network constitutive model. Subsequently, the usage of this model within a nonlinear, upper-level finite element model is straightforward. The Sobolev metrics (e.g., [12; 13]) is also tested in the minimization problem. The proposed neural model allows a prediction of the material response from the strain effects based on data set \(\{\epsilon_{i},\sigma_{i},(\partial\sigma/\partial\epsilon)_{i},\ i=1,2,...,N\}\). Two types of constitutive neural networks are proposed. The first approach involves the training of a residual model with three respective surrogate models for stress components. The second approach considers a separate ANN for each stress component. The first neural model requires less computational cost, since the training of only one residual model is required.