GeONet: a neural operator for learning the Wasserstein geodesic

Optimal transport (OT) offers a versatile framework to compare complex data distributions in a geometrically meaningful way. Traditional methods for computing the Wasserstein distance and geodesic between probability measures require mesh-dependent domain discretization and suffer from the curse-of-dimensionality. We present GeONet, a mesh-invariant deep neural operator network that learns the non-linear mapping from the input pair of initial and terminal distributions to the Wasserstein geodesic connecting the two endpoint distributions. In the offline training stage, GeONet learns the saddle point optimality conditions for the dynamic formulation of the OT problem in the primal and dual spaces that are characterized by a coupled PDE system. The subsequent inference stage is instantaneous and can be deployed for real-time predictions in the online learning setting. We demonstrate that GeONet achieves comparable testing accuracy to the standard OT solvers on simulation examples and the MNIST dataset with considerably reduced inference-stage computational cost by orders of magnitude. Recent years have seen tremendous progress in statistical and computational optimal transport (OT) as a lens to explore machine learning problems. One prominent example is to use the Wasserstein distance to compare data distributions in a geometrically meaningful way, which has found various applications, such as in generative models (Arjovsky et al., 2017), domain adaptation (Courty et al., 2017) and computational geometry (Solomon et al., 2015). Computing the optimal coupling and the optimal transport map (if it exists) can be expressed in a fluid dynamics formulation with the minimum kinetic energy (Benamou and Brenier, 2000).