A differentiable hydrodynamical approach to the Milky Way bar pattern speed with CO longitude--velocity data
We present a differentiable hydrodynamical framework for modeling barred gas flow in the Milky Way and constraining broad low-loss regions in bar-pattern-speed parameter space from Galactic longitude--velocity data. The method evolves a neutral-gas disk in a fixed barred potential, projects it into longitude--velocity ($\ell$--$v$) space, and compares predicted and target maps using a cosine-distance loss applied to processed and masked maps. This emphasizes large-scale morphology rather than the absolute emission scale. Because the forward model is differentiable, gradients with respect to the bar pattern speed can be computed and used for direct optimization in observable space. We validate the method with self-consistency hydrodynamical mocks and with an independent mock generated by a different solver with more realistic interstellar-medium physics. These tests recover or identify low-loss regions near the input pattern speed, showing that the method captures coherent bar-driven structures in $\ell$--$v$ space. We then apply the framework to the observed CO $\ell$--$v$ structure of the inner Milky Way. The data yield broad low-loss regions rather than a unique best-fitting value. These regions include moderate pattern speeds, $|Ω_{\rm b}|\sim30$--$40\,{\rm km\,s^{-1}\,kpc^{-1}}$, consistent with current stellar-dynamical constraints, although their location depends on the gas response time and viewing angle. This first application demonstrates the feasibility of differentiable hydrodynamical modeling of Galactic gas as an independent kinematic test of barred Milky Way models and as a step toward multi-parameter forward modeling in position--position--velocity space.