用于多维随机场条件分布重构的局部Sinkhorn框架

中文摘要 AI 辅助

本文提出一种用于多维随机场条件分布重构的局部Sinkhorn散度框架。通过利用去偏Sinkhorn散度，所提方法构建了可微且计算高效的局部分布匹配目标，用于训练随机神经网络（SNN）。此外，本文为该局部Sinkhorn散度框架建立了理论泛化误差估计，明确刻画了由正则化参数控制的近似偏差与统计效率之间的权衡关系，并揭示了所提局部Sinkhorn散度损失函数如何可高效应用于学习多维随机场模型。该框架为条件分布重构提供了精确局部最优传输的可扩展替代方案，为不确定性量化与概率科学机器学习在几何保真度、统计效率及计算可扩展性之间提供了实用折中。通过多个数值示例，将所提局部Sinkhorn散度框架与其他用于训练SNN的损失函数、其他基于机器学习的不确定性量化框架进行对比，结果表明，所提局部Sinkhorn散度框架在重构精度与计算效率间实现了有效平衡，同时对多维随机系统保持良好的可扩展性。

英文摘要

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.

发表机构

• School of Mathematics, University of Birmingham（伯明翰大学数学学院）
• Department of Mathematics, University of Houston（休斯顿大学数学系）
• Nuffield Department of Medicine, University of Oxford（牛津大学纳菲尔德医学院）

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