GeONet: a neural operator for learning the Wasserstein geodesic
This work addresses the curse-of-dimensionality in optimal transport for researchers and practitioners needing real-time predictions, though it is incremental as it builds on existing neural operator frameworks.
The authors tackled the computational challenge of computing Wasserstein geodesics between probability distributions by introducing GeONet, a neural operator that learns the mapping from endpoint distributions to the geodesic, achieving comparable accuracy to standard solvers with orders of magnitude faster inference.
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.