GeONet: a neural operator for learning the Wasserstein geodesic

TL;DR

GeONet is a neural operator that efficiently learns Wasserstein geodesics for probability distributions, enabling real-time optimal transport computations without domain discretization.

Contribution

We introduce GeONet, a mesh-invariant neural operator that learns the OT geodesic mapping directly, reducing computational costs and enabling real-time predictions.

Findings

Achieves comparable accuracy to standard OT solvers.

Reduces inference computational cost by orders of magnitude.

Works effectively on MNIST dataset.

Abstract

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-specific 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.