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Intra-operative endovascular treatment of intracranial aneurysms calls for rapid and reliable hemodynamic quantification. Angiographic Parametric Imaging (API) is derived from DSA time-density curves (TDCs), yet irregular frame rates and dose-reduction gaps often make conventional pixel-wise Gamma-variate fitting unstable and computationally expensive. We propose PIC-Former, a physics-inspired continuous transformer that reconstructs a physiologically plausible TDC function ĉ(t) from irregular DSA samples. PIC-Former uses Δt-aware causal attention for inter-frame gaps and physics-inspired regularization to encourage valid contrast arrival, peak, and washout dynamics. On 52 paired DSA-4D Flow MRI cases and an independent 11-case CFD cohort, QDSA velocities computed from the reconstructed TDCs achieve relative MAEs of 3.1% and 2.01%, respectively, outperforming Gamma-variate fitting (11.8% on DSA-4D Flow MRI). PIC-Former generates full-field parametric maps for a complete DSA run in 12.45 s and remains stable under a 20% frame-drop stress test. In a retrospective intra-operative risk stratification study, the proposed pipeline improves stratification accuracy from 87.5% to 95.0% compared to the Gamma-variate-based baseline. The code and reproducibility test set is available at https://github.com/LY-SUSTech/PIC-Former.git.