用神经网络解最优传输，无需对抗训练，速度快且保证最优。
Neural Hamilton--Jacobi Characteristic Flows for Optimal Transport
- 基于哈密顿-雅可比方程的特征线方法，直接推导出双向传输映射。
- 单个神经网络最小化损失函数，收敛到最优传输映射，计算量大幅降低。
- 支持多种代价函数和类别条件传输，适用于广泛场景。
我们提出一种基于哈密顿-雅可比（HJ）方程的新框架来求解最优传输（OT）问题，其粘性解唯一刻画了OT映射。通过特征线方法，推导出闭式、双向的传输映射，从而避免数值积分。该方法采用纯最小化框架：仅用一个神经网络，通过从HJ方程特征线导出的损失函数进行训练。此设计保证收敛到最优映射，同时消除对抗训练阶段，显著降低计算复杂度。此外，该框架自然扩展至一类广泛的代价函数，并支持类别条件传输。在多个数据集上的大量实验表明，所提方法具备高精度、可扩展性和高效性，为具有可证明最优性的OT应用提供了一种原理严谨且通用的工具。
原文摘要 · Abstract (English)
We present a novel framework for solving optimal transport (OT) problems based on the Hamilton--Jacobi (HJ) equation, whose viscosity solution uniquely characterizes the OT map. By leveraging the method of characteristics, we derive closed-form, bidirectional transport maps, thereby eliminating the need for numerical integration. The proposed method adopts a pure minimization framework: a single neural network is trained with a loss function derived from the method of characteristics of the HJ equation. This design guarantees convergence to the optimal map while eliminating adversarial training stages, thereby substantially reducing computational complexity. Furthermore, the framework naturally extends to a wide class of cost functions and supports class-conditional transport. Extensive experiments on diverse datasets demonstrate the accuracy, scalability, and efficiency of the proposed method, establishing it as a principled and versatile tool for OT applications with provable optimality.
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