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.

翻译：最优传输理论提供了一种在几何意义上比较复杂数据分布的通用框架。传统计算概率测度间Wasserstein距离与测地线的方法受限于网格依赖的域离散化，且面临维数灾难问题。我们提出GeONet——一种网格无关的深度神经算子网络，该网络学习从初始分布与终端分布输入对到连接两个端点分布的Wasserstein测地线的非线性映射。在离线训练阶段，GeONet学习由耦合偏微分方程组刻画的OT问题动态规划形式在原始空间与对偶空间中的鞍点最优性条件。随后的推理阶段可即时完成，适用于在线学习场景中的实时预测。实验表明，GeONet在仿真示例和MNIST数据集上实现了与标准OT求解器相当的测试精度，同时将推理阶段计算成本降低数个数量级。