Fractional Flow Reserve (FFR), a critical clinical index for quantifying the hemodynamic significance of cerebral arterial stenosis, plays an essential role in non-invasive cerebrovascular diagnosis and treatment planning. However, traditional 3D CFD simulations encounter significant challenges in complex vascular networks, including high computational costs and convergence failures arising from incompatible boundary conditions (BCs). While 1D models offer computational efficiency, they rely on empirical stenosis formulations and exhibit high sensitivity to BCs. To address these limitations, this study presents a two-fold contribution. First, we develop a modified DeepOnet-based surrogate model that learns the 1D pressure-flowrate relationship for individual vessel segments, accommodating both normal and stenotic geometries. Second, we construct a differentiable computational framework for whole-brain FFR calculation, where each vessel segment is modeled by the pre-trained DeepOnet surrogate. Within this framework, unknown boundary conditions are initialized empirically and optimized as learnable parameters via physics-informed loss functions, thereby ensuring self-consistent and converged solutions across the entire vascular network. The framework was validated on 12 cerebral vascular network cases with varying degrees of stenosis. Compared with invasive clinical measurements, the framework achieved a root mean square error (RMSE) of 0.079. Using the clinical threshold of 0.8 for FFR-based diagnosis, the diagnostic accuracy reached nearly 100%, demonstrating robust performance in complex geometries. By eliminating the need for clinicians to manually specify boundary conditions, this framework significantly improves automation and lowers the technical barrier for clinical application.