X-ray and neutron scattering are indispensable for structural characterization across condensed-matter physics and materials science. Yet, because scattering experiments generally record intensities without phase information, distinct structures can yield similar scattering patterns, rendering the corresponding inverse problems non-unique. This ambiguity can be reduced by combining complementary measurements, but their integration into machine-learning frameworks remains challenging. Here we introduce a modular framework for probabilistic multi-measurement analysis based on prior-amortized flow matching. Single-observation posterior estimators are trained separately for individual observation types or physical probes and combined only at inference time through a compositional sampler. This late fusion approach allows pretrained single-observation models to be reused without dedicated joint retraining. As a case in point, we consider reflectometry, which provides depth-resolved, nanometer-scale information about thin films and interfaces. Our single-observation demonstrations span X-ray, time-of-flight neutron and polarized neutron reflectometry. In separate co-refinement studies, task-matched estimators enable homogeneous composition across multiple neutron contrasts and heterogeneous multimodal composition of X-ray and neutron measurements. Together, these results establish compositional flow matching as a modular approach to uncertainty-aware inversion and co-refinement, with potential applicability to other scattering problems in which multiple observations constrain a common physical structure while retaining measurement-specific parameters.

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