A Local Sinkhorn Framework for Conditional Distribution Reconstruction of Multidimensional Random Fields

Reconstructing random fields from observational data is central to uncertainty quantification in physical and engineering systems such as fluid flow and subsurface dynamics. Rather than predicting only conditional expectations, researchers aim to reconstruct entire conditional probability distributions to capture full uncertainty propagation. While optimal transport metrics like the Wasserstein distance provide physically meaningful distribution comparisons, their exact computation requires solving expensive linear programming problems that scale poorly during neural network training. This paper addresses the challenge of making optimal transport-based distribution matching scalable for multidimensional stochastic systems without sacrificing geometric fidelity.

The authors introduce a local Sinkhorn divergence framework to train stochastic neural networks (SNNs) for random field reconstruction. By replacing exact optimal transport with the debiased Sinkhorn divergence, the approach introduces an entropic regularization parameter that makes the objective fully differentiable and amenable to GPU-accelerated matrix scaling algorithms. To handle spatial variability, the method constructs local empirical conditional distributions within spatial neighborhoods of input samples, evaluating the average Sinkhorn divergence across the domain.

The paper establishes rigorous theoretical generalization error bounds connecting the local empirical Sinkhorn divergence to the population-averaged squared Wasserstein discrepancy. These error estimates explicitly characterize the trade-off between approximation bias and statistical efficiency controlled by the regularization parameter and neighborhood size. The analysis demonstrates how the entropic regularization term helps control statistical convergence behavior and mitigate computational bottlenecks associated with high-dimensional empirical measures.