In this work, a decoupled computational homogenization method for nonlinear elastic materials is proposed using Neural Networks (NN). In this method, the effective potential is represented as a response surface parameterized by the macroscopic strains and some microstructural parameters. The discrete values of the effective potential are computed by FEM through random sampling in the parameter space, and NN are used to approximate the surface response and to derive the macroscopic stress and tangent tensor components. We show through several numerical convergence analyses that smooth functions can be efficiently evaluated in parameter spaces with dimension up to 10, allowing to consider 3D Representative Volume Elements (RVE) and an explicit dependence of the effective behavior on microstructural parameters like volume fraction. We present several applications of this technique to the homogenization of nonlinear elastic composites, involving a two-scale example of heterogeneous structure with graded nonlinear properties.