Neural Hamilton-Jacobi Characteristic Flows for Optimal Transport

Neural characteristic flow (NCF) for optimal transport via the Hamilton-Jacobi equation

📄 Paper (ICLR 2026) • 💻 Code

---

📌 Overview

This repository contains the official implementation of the paper:

Neural Hamilton-Jacobi Characteristic Flows for Optimal Transport
ICLR 2026

We introduce a novel framework for solving optimal transport (OT) problems by leveraging the Hamilton-Jacobi (HJ) equation and its method of characteristics.
Unlike existing neural OT approaches, our method derives closed-form, bidirectional transport maps, eliminating the need for numerical integration or adversarial training.

---

✨ Highlights

Our approach is based on the observation that the viscosity solution of the Hamilton--Jacobi equation uniquely characterizes the optimal transport map.

By parameterizing the implicit HJ solution with a neural network and training it using a loss derived from the method of characteristics, we obtain:

• Closed-form transport maps via HJ characteristic flows
• Pure minimization framework (no adversarial training)
• Single neural network with provable convergence
• Bidirectional OT maps without numerical integration
• Class-conditional transport support

---

📂 Repository Structure

.
├── checkpoints/       # Saved model checkpoints
├── data/
│   └── synthetic/     # Synthetic datasets
├── sampler/           # Sampling utilities for data generation
├── utils/             # General utility functions
├── models.py          # Neural Hamilton-Jacobi model definition
├── run.py             # Main training and evaluation script
├── setup.conf         # Configuration file for experiments
├── requirements.txt   # Python package dependencies
└── README.md          # This file

🚀 Getting Started

Installation

git clone https://github.com/Yebbi/NCF.git
cd NCF
pip install -r requirements.txt

Training

To train the NCF model on a standard optimal transport benchmark:

python run.py \
    --home_dir path/to/home \
    --conf path/to/setup.conf \
    --data_dir path/to/data \
    --input input_data.npy \
    --output output_data.npy \
    --gpu gpu_id

---

📎 Citation

If you find this work useful, please consider citing:

@article{park2025neural,
  title={Neural Hamilton--Jacobi Characteristic Flows for Optimal Transport},
  author={Park, Yesom and Liu, Shu and Zhou, Mo and Osher, Stanley},
  journal={International Conference on Learning Representations},
  year={2026},
  url={https://openreview.net/forum?id=YbQxus1KEa}
}