We propose a novel formulation of elastic materials based on high‐order interpolants, which fits accurately complex elastic behaviors, but remains conservative. The proposed high‐order interpolants can be regarded as a high‐dimensional extension of radial basis functions, and they allow the interpolation of derivatives of elastic energy, in particular stress and stiffness. Given the proposed parameterization of elasticity models, we devise an algorithm to find optimal model parameters based on training data. We have tested our methodology for the homogenization of 2D microstructures, and we show that it succeeds to match complex behaviors with high accuracy.