Contemporary deformation-based morphometry offers parametric classes of diffeomorphisms that can be searched to compute the optimal transformation that warps a shape into another, thus defining a similarity metric for shape objects. Extending such classes to capture the geometrical variability in always more varied statistical situations represents an active research topic. This quest for genericity however leads to computationally-intensive estimation problems. Instead, we propose in this work to learn the best-adapted class of diffeomorphisms along with its parametrization, for a shape data set of interest. Optimization is carried out with an auto-encoding variational inference approach, offering in turn a coherent model-estimator pair that we name diffeomorphic auto-encoder. The main contributions are: (i) an original network-based method to construct diffeomorphisms, (ii) a current-splatting layer that allows neural network architectures to process meshes, (iii) illustrations on simulated and real data sets that show differences in the learned statistical distributions of shapes when compared to a standard approach.