Sparse particle tracks can still yield turbulent flow fields: physics-constrained networks degrade more slowly as seeding thins

Reconstructing a continuous flow field from sparse particle tracks is hardest for velocity gradients and pressure, because particles are scattered in both space and time. Earlier reconstructions either averaged particle velocities within small spatial bins, or fitted a B-spline velocity field under continuity and momentum constraints; the former does not enforce the governing equations and suffers from noise and lost small-scale structure under sparse seeding. All three methods used the same TrackFit-smoothed particle positions, velocities and accelerations, on homogeneous isotropic turbulence and a friction-Reynolds-number-1000 turbulent boundary layer, with training particles down-sampled step by step while the reference field stayed fixed.

The physics-informed network maps space and time coordinates directly to velocity components and pressure, and its output is differentiable, so spatial and temporal derivatives come from automatic differentiation. The training loss combines a data term fitting measured particle velocities with physics terms requiring zero divergence and the incompressible Navier-Stokes momentum equations, balanced by manually chosen weights. The network is a fully connected multilayer perceptron with 10 hidden layers of 400 neurons, tanh activation, inputs normalized to [-1,1], and a fixed-random-frequency Fourier feature mapping before the first layer to counter the network's bias toward low frequencies.

Training also adds collocation points sampled from the equation-residual distribution and uses the SOAP optimizer to handle simultaneous optimization of several loss terms. Uniform random collocation treats all regions alike, so regions already satisfying the equations keep being sampled while high-residual regions are under-sampled. In the synthetic homogeneous isotropic turbulence case, residual-adaptive points were added after 2×10⁴ iterations, and the mean equation loss at those locations fell faster than with uniform points alone; SOAP kept higher inter-step gradient alignment through most of training and converged faster in both data and equation losses.

The scope is limited to incompressible flow, and pressure is the only physical quantity inferred beyond measured velocity; the cases are homogeneous isotropic turbulence, a synthetic turbulent boundary layer at friction Reynolds number 1000, and an experimental boundary layer at 10 m/s free-stream velocity and friction Reynolds number 995 in the German Aerospace Center's one-meter wind tunnel in Göttingen. The experimental case has no dense reference velocity, pressure or vorticity field, so its results indicate generalization and robustness to particle sparsity rather than direct validation of the complete reconstructed field. The authors propose future work on jointly reconstructing particle trajectories and Eulerian fields, inferring additional hidden quantities such as temperature, and developing more efficient large-scale training strategies.

Reconstruction relies on TrackFit-preprocessed particle trajectories, which may be incomplete, carry fitting uncertainty, and are not explicitly constrained to satisfy the governing equations. A smaller divergence residual does not mean a more accurate reconstruction: FlowFit drives divergence to near machine precision by hard constraint, while the PINN divergence residual decreases as particles thin even as velocity and vorticity errors rise. The synthetic boundary-layer Reynolds-stress profiles use only 50 time steps, so statistics are not fully converged; the cause of FlowFit's near-wall artifacts in the experimental case would need a dedicated sensitivity study. On cost, PINNs took the longest runtime in every case, and ran on a GPU while the other two methods ran on a CPU.