Spectral Attention Operator for Learning Boundary-Driven Hemodynamics

Patient-specific simulation of boundary-driven hemodynamics requires solving the governing PDEs for every inlet waveform and physical parameter setting. We introduce the Spectral Attention Operator (SAO), which learns the map from an inlet waveform to the response at any queried location and parameter setting. SAO couples temporal Fourier modes according to the inlet and adapts its temporal functions to location and physical parameters through Feature-wise Linear Modulation. It predicts complete trajectories in one forward pass without temporal rollout. Across three viscous Burgers and two hemodynamics benchmarks, SAO attains the best overall accuracy in comparisons with five capacity-matched operator families, with relative $L^2$ errors near or below $1$%. On the multi-frequency and hemodynamic flow-rate tasks, SAO reduces error by $46$-$71$% against the strongest external baseline. Ablations isolate the contributions of cross-mode attention and query conditioning, and physics-based diagnostics show accurate shock localization and hemodynamic fields consistent with the governing relations. For hemodynamics, SAO is hundreds to thousands of times faster than numerical simulation on CPU and four to five orders of magnitude faster on GPU.