  Deep learning (DL) has recently emerged as a candidate for closure modeling
of large-eddy simulation (LES) of turbulent flows. High-fidelity training data
is typically limited: it is computationally costly (or even impossible) to
numerically generate at high Reynolds numbers, while experimental data is also
expensive to produce and might only include sparse/aggregate flow measurements.
Thus, only a relatively small number of geometries and physical regimes will
realistically be included in any training dataset. Limited data can lead to
overfitting and therefore inaccurate predictions for geometries and physical
regimes that are different from the training cases. We develop a new online
training method for deep learning closure models in LES which seeks to address
this challenge. The deep learning closure model is dynamically trained during a
large-eddy simulation (LES) calculation using embedded direct numerical
simulation (DNS) data. That is, in a small subset of the domain, the flow is
computed at DNS resolutions in concert with the LES prediction. The closure
model then adjusts its approximation to the unclosed terms using data from the
embedded DNS. Consequently, the closure model is trained on data from the exact
geometry/physical regime of the prediction at hand. An online optimization
algorithm is developed to dynamically train the deep learning closure model in
the coupled, LES-embedded DNS calculation.
