  Optimal transport has been an essential tool for reconstructing dynamics from
complex data. With the increasingly available multifaceted data, a system can
often be characterized across multiple spaces. Therefore, it is crucial to
maintain coherence in the dynamics across these diverse spaces. To address this
challenge, we introduce Synchronized Optimal Transport (SyncOT), a novel
approach to jointly model dynamics that represent the same system through
multiple spaces. With given correspondence between the spaces, SyncOT minimizes
the aggregated cost of the dynamics induced across all considered spaces. The
problem is discretized into a finite-dimensional convex problem using a
staggered grid. Primal-dual algorithm-based approaches are then developed to
solve the discretized problem. Various numerical experiments demonstrate the
capabilities and properties of SyncOT and validate the effectiveness of the
proposed algorithms.
