Neural Hamilton--Jacobi Characteristic Flows for Optimal Transport
This provides a more efficient and principled tool for researchers and practitioners in machine learning dealing with optimal transport applications, though it is an incremental improvement over existing methods.
The paper tackles the computational complexity of optimal transport problems by introducing a neural network framework based on the Hamilton-Jacobi equation, which eliminates adversarial training and numerical integration, achieving accurate and scalable results across diverse datasets.
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.