论文标题
Geonet：学习Wasserstein Geodesic的神经操作员
GeONet: a neural operator for learning the Wasserstein geodesic
论文作者
论文摘要
Optimal Transport（OT）提供了一种多功能框架，以几何有意义的方式比较复杂的数据分布。计算Wasserstein距离的传统方法和概率措施之间的大地测量需要离散网状域，并且受差异性的诅咒。我们提出了Geonet，这是一种网状不动物的深神经操作员网络，该网络从一对初始和终端分布对连接两个端点分布的Wasserstein Geodesic中学习非线性映射。在离线训练阶段，Geonet了解了以耦合PDE系统为特征的原始和双空间中OT问题动态提出的鞍点最佳条件。随后的推理阶段是瞬时的，可以在在线学习环境中进行实时预测。我们证明，在模拟示例和MNIST数据集上，Geonet与标准OT求解器达到了可比的测试精度，其通过数量级降低了推理阶段计算成本。
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.
