Uncertainty quantification for data-driven turbulence modelling with Mondrian forests

Abstract

Data-driven turbulence modelling approaches are gaining increasing interest from the CFD community. Such approaches generally aim to improve the modelled Reynolds stresses by leveraging data from high fidelity turbulence resolving simulations. However, the introduction of a machine learning (ML) model introduces a new source of uncertainty, the ML model itself. Quantification of this uncertainty is essential since the predictive capability of a data-driven model diminishes when predicting physics not seen during training. In this work, we explore the suitability of Mondrian forests (MF's) for data-driven turbulence modelling. MF's are claimed to possess many of the advantages of the commonly used random forest (RF) machine learning algorithm, whilst offering principled uncertainty estimates. An example test case is constructed, with a turbulence anisotropy constant derived from high fidelity turbulence resolving simulations. A number of flows at several Reynolds numbers are used for training and testing. MF predictions are found to be superior to those obtained from a linear and non-linear eddy viscosity model. Shapley values, borrowed from game theory, are used to interpret the MF predictions. Predictive uncertainty is found to be large in regions where the training data is not representative. Additionally, the MF predictive uncertainty is found to exhibit stronger correlation with predictive errors compared to an a priori statistical distance measure, which indicates it is a better measure of prediction confidence. The MF predictive uncertainty is also found to be better calibrated and less computationally costly than the uncertainty estimated from applying jackknifing to random forest predictions. Finally, Mondrian forests are used to predict the Reynolds discrepancies in a convergent-divergent channel, which are subsequently propagated through a modified CFD solver. The resulting flowfield predictions are in close agreement with the high fidelity data. A procedure for sampling the Mondrian forests' uncertainties is introduced. Propagating these samples enables quantification of the uncertainty in quantities of interest such as velocity or a drag coefficient, due to the uncertainty in the Mondrian forests' predictions. This work suggests that uncertainty quantification can be incorporated into existing data-driven turbulence modelling frameworks by replacing random forests with Mondrian forests. This would also open up the possibility of online learning, whereby new training data could be added without having to retrain the Mondrian forests.