Inductive Biases in Field-Level Cosmological Inference from Galaxy Catalogs

We perform field-level likelihood-free inference of the matter density parameter Omegam from simulated galaxy catalogs using machine learning models with differing inductive biases. Using hydrodynamic simulations from CAMELS, we examine how observable choice and architecture govern cosmological information extraction. We consider galaxy positions and line-of-sight peculiar velocities, separately and jointly, and compare permutation-invariant Deep Sets, implemented with either multilayer perceptrons (MLPs) or Kolmogorov-Arnold Networks (KANs), to graph neural networks (GNNs), which explicitly encode spatial relations. We test in-distribution and out-of-distribution (OOD) performance across simulations with different subgrid galaxy-formation prescriptions. Deep Sets infer Omegam from velocities alone with mean relative errors of approximately 18% in-distribution and ~25% OOD, with KANs and MLPs achieving comparable performance. In contrast, the same set-based approach does not yield useful σ8 predictions in either in-distribution or cross-suite tests. Adding positions does not improve Deep Sets, while GNNs infer Omegam with mean relative errors of about 10% in-distribution and 10--17% OOD. These results indicate that peculiar velocities provide the dominant source of Omegam information for set-based models in this setting, while spatial information is most effectively used by architectures that explicitly encode galaxy-galaxy relations. Because the velocity inputs are exact simulated peculiar velocities, applications to survey data will require validation under realistic velocity-measurement noise, selection effects, and survey geometry.