A data-driven approach to viscous fluid dynamics

In order to describe the behaviour of viscous fluids, two different types of modelling assumptions are typically used: first principles, like the balance of forces, and material specific constitutive laws describing for instance the relation between the strain rate and the stress. Combining both, one obtains the partial differential equations of fluid mechanics like the Navier–Stokes system. This approach generates two errors with respect to modelling the real world: First, measurement errors appear due to the imperfectness of experimental equipment. Second, modelling errors are induced when deriving constitutive laws from experimental data.

In this project, instead of including constitutive laws in the mathematical models, we directly use experimental data to determine pairs of stress and strain rate. We find the pairs that approximate the experimental data best and satisfy the differential constraints coming from first principles. Thus, in contrast to the classical approach, the data-driven approach avoids modelling errors.

Moreover, data-driven solutions provide a new relaxed solution concept. Both in the case of the stationary and in the case of the time-dependent Navier—Stokes system we aim at proving that the constructed data-driven solutions are consistent with solutions to the classical PDEs of fluid mechanics if the data sets have the form of a monotone constitutive relation.