  Reliable training of generative adversarial networks (GANs) typically require
massive datasets in order to model complicated distributions. However, in
several applications, training samples obey invariances that are \textit{a
priori} known; for example, in complex physics simulations, the training data
obey universal laws encoded as well-defined mathematical equations. In this
paper, we propose a new generative modeling approach, InvNet, that can
efficiently model data spaces with known invariances. We devise an adversarial
training algorithm to encode them into data distribution. We validate our
framework in three experimental settings: generating images with fixed motifs;
solving nonlinear partial differential equations (PDEs); and reconstructing
two-phase microstructures with desired statistical properties. We complement
our experiments with several theoretical results.
