The Navier-Stokes equations describes the motion of all fluids around us, *i.e.* being able to accurately solve these equations enables us to predict their future behavior, given their current state. This is relevant for a wide range of topics such as wind farm design, weather forecasting, *etc*. However, for large Reynolds numbers we require increasingly fine spatial grids to sufficiently resolve the relevant physics within a numerical simulation. This massively increases the required computational effort. In the large eddy simulation framework, we circumvent this issue by resolving only part of the physics on a coarse grid, and choose to model the smaller sub-grid scale features (turbulence) by some closure model.

Recently machine learning approaches, and typically neural networks, have been increasingly researched for the construction of such closure models. However, stability of the resulting coupled physics + neural network system remains an open issue. In this project we therefore aim to develop stable machine learning closure models by enforcing known physical constraints, such as energy and momentum conservation, onto the neural network. In this way we aim to achieve stability by preventing an unphysical influx of energy into the system.

For this subproject the PhD student Toby van Gastelen has been appointed under the supervision of lead researcher Benjamin Sanderse. Both are located in the Scientific Computing group at the Centrum Wiskunde & Informatica.