The vast design space of multi-functional metamaterials has motivated the need for inverse design to identify the optimal structural topology that satisfies the requirement of multi-physical properties. However, this challenging problem has not been properly solved due to several primary reasons: (i) The “curse of dimension” problem due to the vast array of structural topologies and material constituents; (ii) Different physical properties are often strongly coupled, even conflicting, which makes it difficult to simultaneously optimise and independently adjust them; (iii) Mapping physical properties to structural topology space is an ill-posed problem, because multiple topologies can have the same effective properties. In this study, a data-driven computational framework is developed for multi-objective inverse design of spinodoid cellular metamaterials with desired anisotropic properties, which enables us to simultaneously tailor the mechanical deformation, fluid transport and heat transfer performances. Cellular structures with various spinodal topologies are generated within the full design space, and multiple physical properties, including mechanical stiffness, hydraulic permeability and thermal conductivity tensors, are numerically evaluated via finite element analysis (FEA) and computational fluid dynamics (CFD) respectively. After having a sufficiently large database, regressional conditional generative adversarial network (rcGAN) is used to construct the one-to-many mapping to represent inverse structure-property relations, from which multiple cellular structures can be generated by inputting the target mechanical-transport-thermal properties. The preliminary results demonstrate that this new computational design framework is an effective tool to deal with stiffness-permeability-diffusivity synergy, expand the tunable scope of multiphysical performances, and tailor spinodoid cellular metamaterials with the desired multi-functionality.