Water in a river flows around rocks, air bends around an airplane wing: these motions are described by equations that are hard to solve even on a powerful computer. Neural networks can approximate such solutions, but they often err when data is collected at only a few points or is noisy. The reason is that the network has a poor sense of the local balance of forces: somewhere the flow accelerates, somewhere it slows down, and all of this must be captured simultaneously.
The authors of the new study proposed adding a learnable parameter to the model, which they called "viscosity modulation." Viscosity is a measure of internal friction in a fluid: honey flows more slowly than water because its viscosity is higher. In the equations of motion, viscosity appears as a multiplier of the term responsible for diffusion (the spreading of momentum). The neural network can now change this multiplier at each point in space and time, adjusting the friction force to the features of the flow. It is as if the river had stretches with different water thickness, and the network learned to tune them.
Experiments on two-dimensional problems (e.g., Kovasznay flow) showed that this approach stabilizes training and improves the accuracy of flow reconstruction from sparse and noisy data. Comparisons with other architectures (GRU, attention) confirmed the method's advantage. This is a step toward neural networks reliably predicting the behavior of liquids and gases in engineering and science.
The river water mentioned at the start can now be modeled more accurately even from fragmentary observations — thanks to viscosity that adapts to the flow itself.
Air viscosity is so low that without accounting for turbulence, airplanes would not take off: lift depends on a thin layer of air near the wing, where friction plays a key role.