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Abstract: Simulating complex physical systems often requires joining non-uniform subsystems, which may be characterized by different geometries or mesh topologies. Coupling these separate subsystems often relies on time-intensive meshing workflows or empirical coupling models, which may not generalize well across all operational regimes. The Schwarz alternating method proposes to overcome these issues, establishing an effective domain decomposition framework that allows for the coupling of arbitrary geometries. This talk presents a brief history of Schwarz-based coupling work at Sandia National Laboratories, along with recent work on combining the Schwarz alternating method with data-driven modeling approaches, namely projection-based reduced order models (PROMs) and operator inference. This approach can generate surrogates that are capable of simulating advection-dominated fluid flows with higher accuracy and lower cost than comparable monolithic models, aiding analysis in many-query applications such as uncertainty quantification and engineering design. Several nuances of the Schwarz algorithm and their impacts on model performance are explored, specifically non-overlapping decompositions and PROM hyper-reduction under domain decomposition. A look into ongoing Schwarz coupling work at Sandia discusses existing challenges and efforts to apply this approach to relevant engineering problems.