One Network Learns the Whole Flow and the Path of a Single Speck
There are two ways to describe moving water, and physicists have used both for more than two centuries. You can stand at a fixed point and record how fast the fluid goes past you; that is the Eulerian description, and it is what almost every flow instrument measures. Or you can pick one parcel of fluid and follow it wherever it goes; that is the Lagrangian description, and it is the one you want when the question is where something ends up.
Getting from the first to the second normally means solving the equations of fluid motion, which is slow and hungry for data. Jingdi Wan, Yang Liu and colleagues at the Institute of Mechanics of the Chinese Academy of Sciences in Beijing have trained a single neural network to deliver both at once. Their model, TrajectoryFlowNet, was published Oct. 20, 2026, in Nature Machine Intelligence. It has two halves trained together. One predicts the velocity everywhere in the flow at every moment. The other predicts the entire path of a particle released at a chosen spot. The two halves are welded by a condition the training enforces: the speed along the predicted path must equal the velocity the field half gives at that place and time.
Two further conditions come from physics rather than from data. The network is penalized when its predicted velocity and pressure fail the momentum balance for a fluid that cannot be compressed, and penalized again when the fluid's volume is not conserved, with as much leaving any small region as enters it. Those are the Navier-Stokes equations, the standard account of how a fluid pushes itself around. They enter the training as penalties to be minimized, not as hard rules. Nothing forces the output to satisfy them exactly, and the authors put the aim as physical consistency rather than conservation.
The part worth noting is that it works from what the paper calls scarce measurement. Scarce is specific here, and it is not the same as small. The shortage is in particle paths followed over many successive frames, and in how much of the flow was sampled at all; what the network gets is hundreds of tracked paths and a few tens of camera snapshots instead of a dense field. The authors tested that sparseness against their own full training set, thinning it down in steps. They do not test whether a competing method would need more data, and they do not claim it: the two methods they compare against were handed the same tracks, plus extra information about the boundaries.
The aorta is silicone, and the fluid is not blood
Two of the four test cases are computer simulations: flow in a square box stirred by a sliding lid, and flow past a cluster of cylinders. The other two are the ones the paper leads with, and they are bench hardware. One is an aorta 3D-printed in silicone from the CT scan of a healthy person, fitted with a replacement heart valve and plumbed into a loop that pulses like a circulation. The other is a silicone left ventricle driven by a piston, with a real excised valve complex sutured into it; the paper names no species for that tissue. Both rigs are seeded with fluorescent tracer specks and filmed with a pulsed laser and a camera, the standard technique known as particle image velocimetry.
The fluid circulating through them is not blood, and the published data say so plainly. Its density is 1,328 kilograms per cubic meter, where human blood is around 1,060. Its viscosity, or thickness, sits close to blood's. That mismatch is deliberate. Urea was dissolved into the water-and-glycerol mixture so that the liquid bends light in the same way as the silicone around it, which is what lets a laser and camera see into the model. Thickness is what shapes the flow pattern; transparency is what makes it measurable.
Where the agreement holds, and where it thins
The paper's numbers split in two, and the split matters. Particle paths come out very well: correlation with the tracks actually measured or simulated stays above 0.9965 in all four cases. The velocity field is a different matter. On the two simulations it is close to perfect; on the aorta and the ventricle, the velocity components agree with the measurements at correlations of roughly 0.73 to 0.92, which is reasonable rather than close. Pressure inside the two rigs is predicted with an uncertainty estimate attached and checked against nothing, because pressure in those models was never measured.
To see what broke first, the authors also thinned their own training data in stages. The velocity field held up. The particle paths, the capability the model is named for, did not: at about a tenth of the training tracks, the path error grew by more than an order of magnitude. The paper's own summary of that test says accuracy was maintained with the reduced dataset, then concedes the asymmetry in the following line. The first half of that sentence is true of the field and not of the paths.
This is not the first attempt at the problem
The paper's survey of earlier work names more than a dozen efforts to put physics into particle velocimetry, including the two it benchmarks against. Those two are AIV, published by Shengze Cai and colleagues in PNAS in 2021, and AIVT, published by Toscano and colleagues in Science Advances in 2025. The comparison is narrower than it looks. It exists only on the two simulated cases, so the aorta and ventricle results stand on their own. Both rivals were rebuilt in-house by this team, with AIVT's physics constraints swapped for the ones used here, and Cai is a co-author of this paper. In the box-and-lid case, AIV reconstructs pressure slightly better than TrajectoryFlowNet does, and the paper says so, which is what makes the rest of the table worth reading.
What the authors say they want to point the method at next is not medicine. Their own list is environmental and industrial: pollutants and microplastics spreading through the ocean, reconstructed from a thin scatter of drifting buoys; surface currents in the Arctic inferred from satellite-tracked ice floes; the recovery of a wake behind an obstacle in a wind tunnel. Each has the shape of the problem the heart models stood in for: a flow you can sample in only a few places, and a question about where something carried by it ends up. The code and all four datasets are published with the paper, on GitHub, so another group can test the claims instead of accepting them. The journal ran a commentary on the paper the same day, by Kai Fukami and Romit Maulik; Fukami was also one of its named peer reviewers, which makes that piece the venue's framing rather than an outside verdict.