Physics-Informed Reconstruction of Sparse Hemodynamic Fields

Physics-informed neural networks (PINNs) embed governing partial differential equations as constraints within the training loss, but naively weighting this constraint from the start of training traps optimization near a trivial zero-field solution, a premature-physics failure mode that has limited their reliability for sparse-data field recovery. We introduce a training methodology combining a three-phase physics-weight curriculum with exponential-moving-average residual scaling, which defers strong physics enforcement until the network has learned an adequate data-driven representation and adaptively balances multiple governing-equation residuals without manual tuning. We study this methodology on the recovery of complete hemodynamic fields-velocity, pressure, and passive-scalar concentration—from sparse observations, using a CFD-simulated flow within a patient-specific three-dimensional intracranial aneurysm geometry derived from CT imaging, complemented by a synthetic helical-pipe flow with a known analytical solution. Against linear interpolation, proper orthogonal decomposition, and an architecturally identical physics-free network, our approach achieves mean relative ${\ell }_2$ error $0.115$ at $10\%$ measurement density - a $4.1\times$, $5.3\times$, and $2.6\times$ improvement respectively - decreasing monotonically to $0.082$ at full density. An architecture-matched ablation isolates a further $1.8\times$ improvement attributable to the embedded governing equations alone, independent of network capacity, while a five-configuration ablation over the curriculum, residual scaling, and gradient clipping confirms the curriculum as the dominant stabilizer. We further probe the method's generalization through robustness to Gaussian measurement noise and sensitivity to spatial acquisition strategy. Throughout, pressure recovery remains directly coupled to the velocity field via the momentum equation, a dynamical consistency unavailable to baselines that fit each field independently. Together, these results show that principled training-schedule design, not architecture alone, is critical to realizing physics-informed learning's promise for sparse biomedical field reconstruction.