Synchronization of Unbalanced Dynamical Optimal Transport across Multiple Spaces

TL;DR

This paper introduces UnSyncOT, a novel framework for synchronizing unbalanced dynamical optimal transport across multiple heterogeneous spaces, with theoretical analysis and efficient computational methods.

Contribution

It develops a unified dynamical model for unbalanced transport across spaces, providing theoretical reductions, structural analysis, and practical algorithms.

Findings

UnSyncOT can be reduced to a single-space problem.

The framework unifies transport and reaction dynamics.

Proposed algorithms are validated for convergence and efficiency.

Abstract

Many biological systems are observed through heterogeneous modalities, requiring transport models that couple dynamics across spaces while allowing mass variation. To address this challenge, we introduce Unbalanced Synchronized Optimal Transport (UnSyncOT), a novel dynamical framework that synchronizes transport-reaction flows between spaces via either geometric embeddings (Monge type) or Markov kernels (Kantorovich type). For both cases we prove that UnSyncOT can be reduced to a single-space problem: the Monge model becomes a Benamou-Brenier problem with a metric-modified kinetic energy, and the Kantorovich model yields a nonlocal action induced by the synchronization operator, both of which fit within a dissipation-distance formulation. We also analyze the pure transport (Wasserstein) and pure reaction (Fisher-Rao) limits and derive structural properties. For the Kantorovich case we propose an approximate UnSyncOT by introducing a Hellinger-Kantorovich based trapezoidal time discretization of the secondary action for efficient computation. Finally we present staggered-grid discretizations and primal-dual solvers, validate the convergence, stability, and efficiency, and demonstrate coherent dynamics reconstructions across spaces.