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Optimal transport (OT) offers a versatile framework to compare complex datadistributions in a geometrically meaningful way. Traditional methods forcomputing the Wasserstein distance and geodesic between probability measuresrequire mesh-dependent domain discretization and suffer from thecurse-of-dimensionality. We present GeONet, a mesh-invariant deep neuraloperator network that learns the non-linear mapping from the input pair ofinitial and terminal distributions to the Wasserstein geodesic connecting thetwo endpoint distributions. In the offline training stage, GeONet learns thesaddle point optimality conditions for the dynamic formulation of the OTproblem in the primal and dual spaces that are characterized by a coupled PDEsystem. The subsequent inference stage is instantaneous and can be deployed forreal-time predictions in the online learning setting. We demonstrate thatGeONet achieves comparable testing accuracy to the standard OT solvers onsimulation examples and the MNIST dataset with considerably reducedinference-stage computational cost by orders of magnitude.

GeONet: a neural operator for learning the Wasserstein geodesic