用神经微分方程逼近非平衡最优传输，提升机器学习中的计算传输基础。
Control, Optimal Transport and Neural Differential Equations in Supervised Learning
- 基于皮尔逊散度构造连续非平衡最优传输的向量场
- 提出类似Sinkhorn算法的数值方案并证明收敛性与误差界
- 构建神经微分方程使流逼近真实传输动态，适用于理论驱动的生成模型
我们研究了使用神经微分方程（Neural ODEs）近似最优传输（OT）方程这一基本计算问题。具体而言，提出一种新框架，在连续情况下利用神经微分方程近似非平衡最优传输（UOT）。通过将离散的非平衡最优传输问题推广至皮尔逊散度形式，构造出收敛于真实UOT动力学的向量场，从而推进了计算传输与机器学习的数学基础。为此，设计了一种受Sinkhorn算法启发的数值方案以求解对应最小化问题，并严格证明其收敛性，给出明确的误差估计。从数值解中导出定义传输动力学的向量场，构建相应的传输方程。最后，基于数值获得的传输方程，构造出神经微分方程，其流在适当极限下收敛至真实的传输动力学。
原文摘要 · Abstract (English)
We study the fundamental computational problem of approximating optimal transport (OT) equations using neural differential equations (Neural ODEs). More specifically, we develop a novel framework for approximating unbalanced optimal transport (UOT) in the continuum using Neural ODEs. By generalizing a discrete UOT problem with Pearson divergence, we constructively design vector fields for Neural ODEs that converge to the true UOT dynamics, thereby advancing the mathematical foundations of computational transport and machine learning. To this end, we design a numerical scheme inspired by the Sinkhorn algorithm to solve the corresponding minimization problem and rigorously prove its convergence, providing explicit error estimates. From the obtained numerical solutions, we derive vector fields defining the transport dynamics and construct the corresponding transport equation. Finally, from the numerically obtained transport equation, we construct a neural differential equation whose flow converges to the true transport dynamics in an appropriate limiting regime.
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