We propose using deep neural networks for the fast retrieval of effective properties of metamaterials based on their angular-dependent reflection and transmission spectra from thin slabs. While we noticed that non-uniqueness is an issue for a successful application, we propose as a solution an automatic algorithm to subdivide the entire parameter space. Then, in each sub-space, the mapping between the optical response (complex reflection and transmission coefficients) and the corresponding material parameters (dielectric permittivity and permeability) is unique. We show that we can easily train one neural network per sub-space. For the final parameter retrieval, predictions from the different sub-networks are compared, and the one with the smallest error expresses the desired effective properties. Our approach allows a significant reduction in run-time, compared to more traditional least-squares fitting. Using deep neural networks to retrieve effective properties of metamaterials is a significant showcase for the application of AI technology to nanophotonic problems. Once trained, the nets can be applied to retrieve properties of a larger number of different metamaterials.