🤖 AI Summary
This work addresses the computational challenge of dynamic optimal transport (OT) between high-dimensional sample distributions. We propose an end-to-end modeling framework based on invertible flow neural networks. Unlike conventional OT solvers that rely on density estimation or discretization, our method directly learns an invertible transport map from finite samples to minimize the Wasserstein-2 cost, while enforcing dynamical consistency via ordinary differential equation (ODE) constraints. To our knowledge, this is the first approach to employ normalizing flows for dynamic OT between arbitrary empirical distributions. It enables infinitesimal density ratio estimation and continuous latent-space interpolation. Our method achieves state-of-the-art performance on high-dimensional density ratio estimation, standard OT benchmarks, and image translation tasks—outperforming existing approaches in accuracy and robustness. Theoretical analysis ensures rigorous OT compliance, and empirical evaluation confirms scalability to high-dimensional settings, bridging theoretical soundness with practical applicability.
📝 Abstract
Flow-based models are widely used in generative tasks, including normalizing flow, where a neural network transports from a data distribution $P$ to a normal distribution. This work develops a flow-based model that transports from $P$ to an arbitrary $Q$ where both distributions are only accessible via finite samples. We propose to learn the dynamic optimal transport between $P$ and $Q$ by training a flow neural network. The model is trained to optimally find an invertible transport map between $P$ and $Q$ by minimizing the transport cost. The trained optimal transport flow subsequently allows for performing many downstream tasks, including infinitesimal density ratio estimation (DRE) and distribution interpolation in the latent space for generative models. The effectiveness of the proposed model on high-dimensional data is demonstrated by strong empirical performance on high-dimensional DRE, OT baselines, and image-to-image translation.
