Data-driven Bayesian inference of turbulence model closure coefficients incorporating epistemic uncertainty

Abstract

We introduce a framework for statistical inference of the closure coefficients using machine learning methods. The objective of this framework is to quantify the epistemic uncertainty associated with the closure model by using experimental data via Bayesian statistics. The framework is tailored towards cases for which a limited amount of experimental data is available. It consists of two components. First, by treating all latent variables (non-observed variables) in the model as stochastic variables, all sources of uncertainty of the probabilistic closure model are quantified by a fully Bayesian approach. The probabilistic model is defined to consist of the closure coefficients as parameters and other parameters incorporating noise. Then, the uncertainty associated with the closure coefficients is extracted from the overall uncertainty by considering the noise being zero. The overall uncertainty is rigorously evaluated by using Markov-Chain Monte Carlo sampling assisted by surrogate models. We apply the framework to the Spalart–Allmars one-equation turbulence model. Two test cases are considered, including an industrially relevant full aircraft model at transonic flow conditions, the Airbus XRF1. Eventually, we demonstrate that epistemic uncertainties in the closure coefficients result into uncertainties in flow quantities of interest which are prominent around, and downstream, of the shock occurring over the XRF1 wing. This data-driven approach could help to enhance the predictive capabilities of computational fluid dynamics (CFD) in terms of reliable turbulence modeling at extremes of the flight envelope if measured data is available, which is important in the context of robust design and towards virtual aircraft certification. The plentiful amount of information about the uncertainties could also assist when it comes to estimating the influence of the measured data on the inferred model coefficients. Finally, the developed framework is flexible and can be applied to different test cases and to various turbulence models.

Crack localization in glass fiber composite beams by experimental modal analysis and multi variable Gaussian process regression method

“…Machine learning methods have gained popularity for making predictions based on trained experimental data. Maruyama et al. (2021) introduced machine learning methods for statistically inferring closure coefficients, utilizing experimental data via Bayesian statistics to quantify associated epistemic uncertainty.…”

“…Machine learning methods have gained popularity for making predictions based on trained experimental data. Maruyama et al (2021) introduced machine learning methods for statistically inferring closure coefficients, utilizing experimental data via Bayesian statistics to quantify associated epistemic uncertainty. Wang (2022) provided a comprehensive tutorial on Gaussian process regression in machine learning applications, covering fundamental concepts such as kernels and conditional probability.…”

Learning to Simulate Fluid Flow Around Airfoils Using Graph Neural Networks

“…This notion aligns with recent developments in PINNs and their Bayesian extensions. Maruyama et al [22] contributed to this field by utilizing Bayesian statistics to infer turbulence model closure coefficients as well as the associated epistemic uncertainty from data. Their work demonstrates the potential of combining data-driven approaches, physical knowledge, and uncertainty quantification to advance the state of turbulence modeling.…”