A-priori energy-biased training for the prediction of closure terms of the VMS Navier-Stokes equations

The authors present an energy-biased training approach for predicting exact closure terms of the Navier-Stokes equations discretised in a Variational Multiscale framework. The approach initially involves formulating a constrained objective function, which is transformed into an unconstrained problem suitable for neural network training using the Augmented Lagrangian method. The constraints induce a prediction bias by enforcing the predicted closure term energy contributions to be smaller than the exact values, both for excessive backscatter or insufficient dissipation. Effectively, the approach controls energy contributions through a-priori measures rather than a-posteriori measures. The approach was applied to obtain the H¹ projection of a Re_τ = 180 turbulent channel flow on a 32 × 32 × 32 uniform mesh. Eight MLPs were trained to predict the closure terms associated with the weighting functions of each linear hexahedral element. Each network used 217 features, and five hidden layers with 600 neurons each. The final data-to-parameter ratio was ∼19.6: 1 (30 932 992: 1 575 604) per neural network. A-priori evaluation of the networks’ outputs demonstrated its ability to predict closure terms yielding the desired behaviour in energy transfer. This was true for both closure terms that yield energy gain and energy loss. In spite of the energy bias, the closure term predictions retained correlations greater than 0.85 with their exact value for all positions between the channel walls.