Unpaired measure transport with dynamical alignment

For decades, numerical models have been central to forecasting and understand- ing physical systems. Yet unresolved scales, simplified physics, and uncertain parameters make them imperfect. Over long trajectories, model states and ob- servations therefore lie on different attractors, described by different invariant measures. Relating these measures is a transport problem for which generative models provide expressive maps. Standard transport objectives, however, rely on geometric proximity and may pair states whose futures are dynamically incompati- ble. We introduce a framework for transporting invariant measures while aligning their dynamics. We define a discrepancy that quantifies how well two systems can be aligned up to an admissible change of coordinates. We then bound the map-dependent commutation residual by a pairwise ground cost independent of the transport map, reducing dynamical alignment to a linear optimal transport problem. From unpaired trajectories, intertwining Extended Dynamic Mode Decomposition (iEDMD) learns this cost by aligning reduced representations of the two dynam- ics. The resulting coupling supervises a Flow Matching model without requiring integration of the transport during training. The learned map can then be wrapped around an unchanged numerical solver in a transport–solve–transport procedure. Experiments on chaotic systems and fluid flows show that the dynamical cost improves conjugation when geometry and dynamics disagree; when the systems already share their coordinates, geometric costs or simpler maps remain preferable.