A data-driven computational framework is established to implement surrogate constitutive models for porous elastomers undergoing large deformation. Explicit finite element (FE) simulations are conducted to compute the homogenised response of a cubic unit cell of a porous compressible elastomer, subject to a random set of imposed multiaxial strain states. The FE predictions are used to assemble a training dataset for two different surrogate models, based on simple neural networks. The first establishes a non-linear correspondence between six-dimensional strain and stress vectors; the second provides a strain energy potential from which to derive the stress versus strain response. The accuracy of the surrogate models is quantified, and their predictions are compared to those of the Hyperfoam model; it is found that the surrogate models can significantly outperform this well-known phenomenological model.