The disorder-driven superconductor-insulator transition is a long standing and paradigmatic example of a quantum phase transition arising from the interplay of strong disorder and interactions. Here we investigate this problem in the two-dimensional disordered attractive Hubbard model using Neural Quantum States, an advanced variational approach that enables accurate computation of ground state properties beyond the limitations of Determinant Quantum Monte Carlo and Bogoliubov–de Gennes mean field theory. We demonstrate that Neural Quantum States, with an appropriately designed variational ansatz is capable of capturing superconducting correlations and efficiently describes the superconductor-insulator transition. Our results reveal the emergence of spatial granularity consistent with earlier mean-field studies and indicate that the transition is driven by the loss of global phase coherence due to a vanishing superfluid stiffness. By simultaneously accounting for both amplitude and phase fluctuations, we find a direct link between island formation and local currents, providing new insight into this phenomenon.