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Physics-Guided Machine Learning for Uncertainty Quantification in Turbulence Models

Minghan Chu, Weicheng Qian

TL;DR

This work tackles the epistemic uncertainty inherent in turbulence modeling by augmenting the Eigenspace Perturbation Method (EPM) with a data-driven modulation of perturbation magnitudes. A lightweight 1D CNN learns the mapping from the modeled turbulence kinetic energy $k^{\mathrm{RANS}}$ to the high-fidelity $k^{\mathrm{DNS}}$, and the learned correction is used to scale EPM perturbations while preserving the Reynolds-stress anisotropy and realizability. Integrated into the EPM framework, the CNN-driven correction yields $R_{ij}^{\mathrm{corr}} = 2\, \hat{k}^{\mathrm{DNS}}\, b_{ij}^{\mathrm{RANS}}$, improving calibration of model-form uncertainty. Validated on SD7003 and periodic-hill cases with paired DNS–RANS data, the approach achieves one-to-two orders of magnitude reduction in error relative to baseline RANS, offering a physically consistent, calibrated uncertainty quantification pathway for turbulent flow predictions.

Abstract

Predicting the evolution of turbulent flows is central across science and engineering. Most studies rely on simulations with turbulence models, whose empirical simplifications introduce epistemic uncertainty. The Eigenspace Perturbation Method (EPM) is a widely used physics-based approach to quantify model-form uncertainty, but being purely physics-based it can overpredict uncertainty bounds. We propose a convolutional neural network (CNN)-based modulation of EPM perturbation magnitudes to improve calibration while preserving physical consistency. Across canonical cases, the hybrid ML-EPM framework yields substantially tighter, better-calibrated uncertainty estimates than baseline EPM alone.

Physics-Guided Machine Learning for Uncertainty Quantification in Turbulence Models

TL;DR

This work tackles the epistemic uncertainty inherent in turbulence modeling by augmenting the Eigenspace Perturbation Method (EPM) with a data-driven modulation of perturbation magnitudes. A lightweight 1D CNN learns the mapping from the modeled turbulence kinetic energy to the high-fidelity , and the learned correction is used to scale EPM perturbations while preserving the Reynolds-stress anisotropy and realizability. Integrated into the EPM framework, the CNN-driven correction yields , improving calibration of model-form uncertainty. Validated on SD7003 and periodic-hill cases with paired DNS–RANS data, the approach achieves one-to-two orders of magnitude reduction in error relative to baseline RANS, offering a physically consistent, calibrated uncertainty quantification pathway for turbulent flow predictions.

Abstract

Predicting the evolution of turbulent flows is central across science and engineering. Most studies rely on simulations with turbulence models, whose empirical simplifications introduce epistemic uncertainty. The Eigenspace Perturbation Method (EPM) is a widely used physics-based approach to quantify model-form uncertainty, but being purely physics-based it can overpredict uncertainty bounds. We propose a convolutional neural network (CNN)-based modulation of EPM perturbation magnitudes to improve calibration while preserving physical consistency. Across canonical cases, the hybrid ML-EPM framework yields substantially tighter, better-calibrated uncertainty estimates than baseline EPM alone.

Paper Structure

This paper contains 7 sections, 9 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Methodology and theoretical background
  3. Neural network architecture and training
  4. Integration into the EPM Framework
  5. Results
  6. Validation, limitations, and discussion.
  7. Conclusions

Figures (3)

  • Figure 1: Data flow and methodology of the proposed framework. The blue path denotes the training stage using paired RANS–DNS data, and the red path indicates the validation and testing stage.
  • Figure 2: Illustration for the SD7003 airfoil case. The first row compares CNN predictions, baseline RANS, and DNS data. The second row shows the MAE of CNN-corrected and baseline predictions at different chordwise positions.
  • Figure 3: Results for turbulent flow over periodic hills: Top—CNN, RANS, and DNS; Bottom—MAE for CNN vs. RANS at multiple $x/h$ stations (streamwise position normalized by hill height).