  A key problem in computational material science deals with understanding the
effect of material distribution (i.e., microstructure) on material performance.
The challenge is to synthesize microstructures, given a finite number of
microstructure images, and/or some physical invariances that the microstructure
exhibits. Conventional approaches are based on stochastic optimization and are
computationally intensive. We introduce three generative models for the fast
synthesis of binary microstructure images. The first model is a WGAN model that
uses a finite number of training images to synthesize new microstructures that
weakly satisfy the physical invariances respected by the original data. The
second model explicitly enforces known physical invariances by replacing the
traditional discriminator in a GAN with an invariance checker. Our third model
combines the first two models to reconstruct microstructures that respect both
explicit physics invariances as well as implicit constraints learned from the
image data. We illustrate these models by reconstructing two-phase
microstructures that exhibit coarsening behavior. The trained models also
exhibit interesting latent variable interpolation behavior, and the results
indicate considerable promise for enforcing user-defined physics constraints
during microstructure synthesis.
