[Paper Review] A Multi-Model Approach for Uncertainty Propagation and Model Calibration in CFD Applications

Jianxun Wang, Christopher J. Roy+1 more|arXiv (Cornell University)|Jan 13, 2015

Probabilistic and Robust Engineering Design26 references3 citations

TL;DR

This paper proposes a multi-fidelity Bayesian approach that combines high- and low-fidelity CFD models to quantify and reduce model discrepancy in uncertainty propagation. By using Gaussian process regression to infer discrepancy functions and Bayesian inference to estimate uncertainties in untested parameter regions, the method improves prediction accuracy beyond what either model can achieve alone.

ABSTRACT

Proper quantification and propagation of uncertainties in computational simulations are of critical importance. This issue is especially challenging for CFD applications. A particular obstacle for uncertainty quantifications in CFD problems is the large model discrepancies associated with the CFD models used for uncertainty propagation. Neglecting or improperly representing the model discrepancies leads to inaccurate and distorted uncertainty distribution for the Quantities of Interest. High-fidelity models, being accurate yet expensive, can accommodate only a small ensemble of simulations and thus lead to large interpolation errors and/or sampling errors; low-fidelity models can propagate a large ensemble, but can introduce large modeling errors. In this work, we propose a multi-model strategy to account for the influences of model discrepancies in uncertainty propagation and to reduce their impact on the predictions. Specifically, we take advantage of CFD models of multiple fidelities to estimate the model discrepancies associated with the lower-fidelity model in the parameter space. A Gaussian process is adopted to construct the model discrepancy function, and a Bayesian approach is used to infer the discrepancies and corresponding uncertainties in the regions of the parameter space where the high-fidelity simulations are not performed. The proposed multi-model strategy combines information from models with different fidelities and computational costs, and is of particular relevance for CFD applications, where a hierarchy of models with a wide range of complexities exists. Several examples of relevance to CFD applications are performed to demonstrate the merits of the proposed strategy. Simulation results suggest that, by combining low- and high-fidelity models, the proposed approach produces better results than what either model can achieve individually.

Motivation & Objective

• Address the challenge of model discrepancy in CFD simulations, which distorts uncertainty propagation when high-fidelity models are too expensive for large-scale sampling.
• Overcome the trade-off between computational cost and accuracy by integrating low-fidelity models (cheap but inaccurate) with high-fidelity models (expensive but accurate).
• Develop a statistical framework to estimate model discrepancies and their uncertainties in regions of the parameter space where high-fidelity simulations are unavailable.
• Improve predictive accuracy and reliability in CFD applications by fusing information from multiple models of varying fidelity.

Proposed method

• Use a hierarchy of CFD models with varying fidelities—high-fidelity for accuracy, low-fidelity for computational efficiency.
• Model the discrepancy between low- and high-fidelity models as a Gaussian process function over the parameter space.
• Apply Bayesian inference to estimate the discrepancy function and quantify its uncertainty in untested regions of the parameter space.
• Leverage sparse high-fidelity simulations to train the discrepancy model and extend predictions to the full parameter domain.
• Combine the low-fidelity model prediction with the inferred discrepancy to produce a corrected, high-accuracy prediction with uncertainty quantification.
• Use the full posterior distribution to propagate uncertainties through the system, ensuring robustness and reliability in predictions.

Experimental results

Research questions

• RQ1 How can model discrepancies between low- and high-fidelity CFD models be effectively quantified and propagated in uncertainty analysis?
• RQ2 To what extent can a Bayesian Gaussian process model improve uncertainty quantification when high-fidelity simulations are limited in number?
• RQ3 Can the integration of multiple models of varying fidelity reduce overall prediction error compared to using a single model?
• RQ4 How does the proposed method perform in capturing the true uncertainty distribution of quantities of interest in CFD applications?

Key findings

• The proposed multi-model strategy significantly reduces prediction errors compared to using only low- or high-fidelity models individually.
• The Bayesian Gaussian process model effectively captures model discrepancies across the parameter space, even in regions without high-fidelity data.
• The method provides more accurate and reliable uncertainty distributions for quantities of interest than single-model approaches.
• By combining sparse high-fidelity data with a discrepancy model, the approach achieves high accuracy at a reduced computational cost.
• The results demonstrate improved robustness and fidelity in uncertainty propagation, especially in regions with limited high-fidelity simulation coverage.