AIR FORCE. The region around a moving wing is physically complex, with strong vortices and steep gradients. Photo credit: Oleh_Slobodeniuk
Pressure is the most important quantity in fluid mechanics and one of the most difficult to measure. Engineers can track the velocity of a flow and follow tracer particles with lasers. But the pressure field, which ultimately determines the forces acting on the wings, turbines and swimming animals, remains largely invisible. Engineers designing small drones that mimic insect flight, or biologists trying to understand how a dragonfly generates lift with each wing, need that data. Most of the time they have to guess it, model it or do without it.
A few years ago, a class of artificial intelligence models called physics-based neural networks (PINN) offered a different approach. Instead of fitting a curve to the data, PINNs incorporate the governing equations of fluid mechanics directly into the learning process. Feed velocity measurements into the model, encode the laws of motion, and the pressure field emerges as a byproduct, inferred rather than measured. The approach lies at the heart of what researchers now call AI for science, a broader movement that includes digital twins of physical systems, where AI learns from known governing laws rather than data alone. Its engineering appeal is straightforward: Instead of running expensive computational fluid dynamics simulations, researchers can recover hidden quantities directly from measured data.
The practical reality, however, was more confusing. The PINNs turned out to be temperamental. They worked well in short time windows and simple flows, but they were asked to follow a system over many cycles of movement (for example, a wing flapping twenty strokes) and the results deteriorated severely. Accumulated errors. Frequencies were lost. The physics got lost somewhere in the math of training. The knee-jerk solution (adding more computing power to the problem) didn’t work: increasing the network size five times over long time domains didn’t produce any significant improvement. To study the kind of complex, long-lived flows that matter most in biology and engineering, standard PINNs were falling short.
Systematic solution
A research team from IIT-Madras and LISN-CNRS laboratory in France has published a systematic solution to this problem. The researchers identified three distinct reasons why PINNs struggle over time: data may be too sparse; the time window is too long; or the flow is too spectrally complex, containing multiple interacting frequencies that no one told the model to look for.
The testbed was an elliptical flapping airfoil operating under conditions typical of insect wings and small unmanned aerial vehicles. The researchers ran two scenarios: periodic flow, repeating with each stroke; and quasi-periodic flow, which is seemingly regular but contains subtle, conflicting frequencies caused by the way air swirls at the leading and trailing edges of the wing at slightly different rates. Quasi-periodic flow is associated with increased lift generation.
The central proposal was to stop treating time as a single and undivided domain. Instead of training a large neural network over the entire history, they divided the temporal domain into segments of two or three flapping cycles each and trained a smaller network on each segment in sequence. At the beginning of each new segment, the network was not initialized from scratch but from the weights of the previously trained network. This is transfer learning: the model continues what it has already learned about the physics and flow structure from the previous interval.
The improvement was substantial: Pressure reconstruction errors fell from 36 percent to around 7 percent. For quasi-periodic flows, the model successfully reconstructed the complex frequency spectrum, including multiple interacting peaks in the entrainment signal, which the standard model completely missed.
The researchers also identified a simpler variant that trains each subsequent segment with fewer iterations and a lower learning rate. It matched the accuracy of the full approach while reducing the training effort by about a third, which is useful for longer stories or more complex geometries.
The team also introduced a practical data strategy they call “preferential spatiotemporal sampling.” The region immediately around the moving wing is physically complex, with strong vortices and steep gradients; the downstream wake is smoother and more predictable. The method concentrates its sampling budget on the chaotic wing interface, resulting in fewer data points, lower computational overhead, and higher accuracy—a significant reduction in GPU time and cloud computing costs.
The immediate application is in experimental fluid mechanics. Take velocity data from a wind or water tunnel, pass it through a trained PINN, and retrieve the pressure field and aerodynamic loads without any additional instrumentation. For bioinspired flight research, where attaching pressure sensors to a dragonfly is not a realistic option, this is an important step. For engineers working on micro-aerial vehicles, small surveillance drones, and search and rescue platforms, the ability to accurately model quasi-periodic flapping over long flight paths is directly relevant to understanding how wing geometry and stroke patterns generate lift.
Limitations
There are limits. Strongly aperiodic or chaotic flows remain out of reach: when the frequency content is wild and the system is sensitive to initial conditions, neural networks lack the representational capacity to keep up. The paper also points out a more subtle restriction: because the training data and pressure benchmarks were produced by two different computational solvers, a small portion of the reported error reflects a disagreement between the tools rather than any weakness in the method itself. And the study was carried out in two dimensions; extending it to realistic 3D wing geometries will require more work in sampling and computational cost.
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Published March 9, 2026