# Flow Field Reconstruction

Starting from noisy velocity images, unsteady three-dimensional blood flow and elastic vessel-wall motion can be reconstructed together without a prescribed inlet waveform or initial velocity. The solver alternates a fluid update, in which a physics-informed network fits the observations while enforcing the incompressible Navier–Stokes equations on the moving domain, with a geometry update, in which registration plus interface traction continuity and an elastic momentum residual constrain wall displacement. On a converging channel and a patient-like aorta, the PirateNets version reaches velocity relative errors of 0.0243 and 0.0372, below the 0.1066 and 0.1305 of a standard PINN of the same depth and width.

Starting from noisy velocity images, unsteady three-dimensional blood flow and elastic vessel-wall motion can be reconstructed together without a prescribed inlet waveform or initial velocity. The solver alternates a fluid update, in which a physics-informed network fits the observations while enforcing the incompressible Navier–Stokes equations on the moving domain, with a geometry update, in which registration plus interface traction continuity and an elastic momentum residual constrain wall displacement. On a converging channel and a patient-like aorta, the PirateNets version reaches velocity relative errors of 0.0243 and 0.0372, below the 0.1066 and 0.1305 of a standard PINN of the same depth and width.

Particle transport in complex flows is usually obtained by solving the Navier–Stokes equations, and numerical or experimental routes demand deep physical insight and high computational cost, while existing machine-learning alternatives fail to generalize long-range particle trajectories under sparse data. TrajectoryFlowNet couples a Lagrangian view of particle tracks with a Eulerian view of full velocity and pressure fields inside one physics-constrained network, so a few trajectory measurements yield both the full-field flow and long-range paths for arbitrary particles. It is validated on numerical lid-driven cavity and complex cylinder flows and on experimental aortic and ventricle blood flows, keeping predictions physically consistent in flows with irregular boundaries.

Particle transport in complex flows is usually obtained by solving the Navier–Stokes equations, and numerical or experimental routes demand deep physical insight and high computational cost, while existing machine-learning alternatives fail to generalize long-range particle trajectories under sparse data. TrajectoryFlowNet couples a Lagrangian view of particle tracks with a Eulerian view of full velocity and pressure fields inside one physics-constrained network, so a few trajectory measurements yield both the full-field flow and long-range paths for arbitrary particles. It is validated on numerical lid-driven cavity and complex cylinder flows and on experimental aortic and ventricle blood flows, keeping predictions physically consistent in flows with irregular boundaries.