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 potential from which to derive the stress versus strain response. The accuracy of the surrogate models is quantified, and its predictions are compared to those of the Hyperfoam model; it is found that the surrogate models can significantly outperform this well-known phenomenological model.