Recent developments (e.g., [1,2]) proposed a three-step homogenization-based approach to topology optimization of lattice structures. One of the main aspects of such an approach is that the minimization problem is written in terms of a relaxed cost function, which holds for a homogenized problem. Thus, a crucial step is computing (off-line) homogenized properties for a given parametrized class of composite materials, which is the main aspect studied in this work. In elasticity and for a given discrete sample of parameters, the homogenized Hooke’s tensor is obtained in terms of the so-called displacement correctors which are solutions of the cell problems. This fourth-order tensor is then used as an input of the homogenized optimization problem (see, for instance, [3] for applications in three- dimensional elasticity). This contribution extends the discussion of two-scale homogenization to the case of square lattice infills with hyperelastic behavior. The main difference is that since the behavior is nonlinear, the homogenized tangent operator depends on the cell parameters, the strains, and, eventually, the number of cells used [4] in the numerical homogenization. Direct computation of all sets of parameters for all strain levels (direction and intensity) becomes prohibitive in this case. Therefore, we propose to train a surrogate Neural Networks (NN) model to predict the set of homogenized tangent operators. Different models are considered: direct training of tensor coefficients, training of invariants based on a polar decomposition, and training of the associated free energy. Finally, we evaluate the predictions of the NN models in well-known instability situations at the micro and macro levels, which are mainly related to buckling and the apparition of shear bands. [1] J. P. Groen, O. Sigmund. Homogenization-based topology optimization for high-resolution manufacturable micro-structures. International Journal for Numerical Methods in Engineering, 2018. [2] G. Allaire, P. Geoffroy-Donders, O. Pantz. Topology optimization of modulated and oriented periodic microstructures by the homogenization method, 2019. [3] P. Geoffroy-Donders, G. Allaire, O. Pantz. 3-d topology optimization of modulated and oriented periodic microstructures by the homogenization method, 2020. [4] S. Müller. Homogenization of nonconvex integral functionals and cellular elastic materials, 1987.