Neural Hamilton--Jacobi Characteristic Flows for Optimal Transport

2025-09-30

AI Summary

This work addresses the optimal transport (OT) problem by proposing a novel neural network framework grounded in the Hamilton–Jacobi (HJ) equation. Methodologically, it pioneers the integration of viscosity solution theory for the HJ equation with the method of characteristics to derive closed-form, bidirectional transport maps—thereby eliminating the need for numerical integration and adversarial training. A physics-informed, end-to-end loss function is designed to jointly optimize a single network, ensuring convergence to the global optimum. Key contributions include: (i) unified support for diverse cost functions—including squared Euclidean and cosine distances—and conditional OT across classes; (ii) theoretical guarantees of optimality alongside computational efficiency. Experiments on multiple benchmark datasets demonstrate substantial reductions in computational complexity while maintaining high accuracy and strong scalability.

Technology Category

• Computer Vision: Learning & Optimization for CV
• Search and Optimization: Learning to Search
• Machine Learning: Optimization

Application Category

• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems
• Graph Algorithms and Modeling for the Web: Graph neural networks and deep learning approaches for Web-related graphs
• Systems and Infrastructure for Web, Mobile and WoT: Experiences and lessons learnt from Web-based algorithms and system deployments

Abstract

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.

Problem

• Solving optimal transport problems using Hamilton-Jacobi equations
• Deriving closed-form bidirectional maps without numerical integration
• Providing scalable optimal transport with provable convergence guarantees

Innovation

• Neural network solves Hamilton-Jacobi equation
• Closed-form bidirectional maps avoid numerical integration
• Single network eliminates adversarial training complexity