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This paper introduces a local Sinkhorn divergence framework aimed at reconstructing conditional distributions of multidimensional random fields through a differentiable and computationally efficient local distribution matching objective. By leveraging the debiased Sinkhorn divergence, the authors establish theoretical generalization error estimates that clarify the trade-offs between approximation bias and statistical efficiency. The proposed method outperforms existing loss functions in training stochastic neural networks, achieving a balance between accuracy and computational scalability for uncertainty quantification in complex systems.
Achieving a scalable and efficient method for conditional distribution reconstruction could revolutionize how we approach uncertainty quantification in multidimensional stochastic systems.
In this paper, we propose a local Sinkhorn divergence framework for conditional distribution reconstruction of multidimensional random fields. By utilizing the debiased Sinkhorn divergence, our proposed approach develops a differentiable and computationally efficient local distribution matching objective to train stochastic neural networks (SNNs). Furthermore, we establish theoretical generalization error estimates for our local Sinkhorn divergence framework, which explicitly characterizes the trade-off between approximation bias and statistical efficiency controlled by the regularization parameter and reveals how our proposed local Sinkhorn divergence loss function can be efficiently applied to learning multidimensional random field models. The proposed framework provides a scalable alternative to exact local optimal transport for conditional distribution reconstruction, offering a practical compromise between geometric fidelity, statistical efficiency, and computational scalability for uncertainty quantification and probabilistic scientific machine learning. Through various numerical examples, we compare our proposed local Sinkhorn divergence framework with other loss functions to train SNNs and with other machine-learning-based uncertainty quantification frameworks, demonstrating that the proposed local Sinkhorn divergence framework achieves an effective balance between reconstruction accuracy and computational efficiency while maintaining good scalability for multidimensional stochastic systems.
