Neural Adaptive Optimal Transport: Duality, Brenier Potentials and Applications

Classical optimal transport is a cornerstone of modern machine learning, yet its rigid requirement for full-mass or fixed-mass conservation severely limits its robustness against outliers and distributional shifts. While recent discrete elastic optimal transport enables adaptive mass transport via inequality constraints and mixed-sign costs, it lacks a continuous theoretical foundation, leaving critical questions on duality, Brenier potentials, and high-dimensional solvability unanswered. To bridge this gap, we propose Neural Adaptive Optimal Transport (NAOT), a novel framework that unifies discrete elastic optimal transport with continuous optimal transport theory. Building upon Brenier's seminal theory, we derive the duality theory and Brenier-type theorems, which shed light on designing an efficient and scalable algorithm based on input convex neural networks to find transport map. NAOT outperforms the state-of-the-art methods at predicting single-cell perturbation responses on the benchmark dataset, demonstrating its superiority on adaptive distribution alignment. Our work provides the first continuous framework of adaptive optimal transport, offering both theoretical insights and a scalable algorithm with broad applicability in artificial intelligence.