  Ultra-cold Fermi gases display diverse quantum mechanical properties,
including the transition from a fermionic superfluid BCS state to a bosonic
superfluid BEC state, which can be probed experimentally with high precision.
However, the theoretical description of these properties is challenging due to
the onset of strong pairing correlations and the non-perturbative nature of the
interaction among the constituent particles. This work introduces a novel
Pfaffian-Jastrow neural-network quantum state that includes backflow
transformation based on message-passing architecture to efficiently encode
pairing, and other quantum mechanical correlations. Our approach offers
substantial improvements over comparable ans\"atze constructed within the
Slater-Jastrow framework and outperforms state-of-the-art diffusion Monte Carlo
methods, as indicated by our lower ground-state energies. We observe the
emergence of strong pairing correlations through the opposite-spin pair
distribution functions. Moreover, we demonstrate that transfer learning
stabilizes and accelerates the training of the neural-network wave function,
enabling the exploration of the BCS-BEC crossover region near unitarity. Our
findings suggest that neural-network quantum states provide a promising
strategy for studying ultra-cold Fermi gases.
