What are the best methods and metrics for quantifying uncertainty in CFD models using machine learning?

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Computational fluid dynamics (CFD) models are widely used to simulate complex fluid phenomena, such as turbulence, combustion, multiphase flows, and aerodynamics. However, CFD models are often affected by various sources of uncertainty, such as numerical errors, model inadequacy, parameter variability, and measurement noise. Quantifying and reducing uncertainty in CFD models is essential for enhancing their reliability, accuracy, and robustness. In this article, you will learn about some of the best methods and metrics for quantifying uncertainty in CFD models using machine learning.

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1 Machine learning for uncertainty quantification

Machine learning (ML) is a branch of artificial intelligence that enables computers to learn from data and perform tasks that are difficult or impossible to solve analytically. ML can be applied to uncertainty quantification (UQ) in CFD models in several ways, such as creating fast and accurate approximations of expensive CFD simulations, learning the difference between CFD models and experimental or high-fidelity data, and directly estimating the uncertainty in CFD model outputs. These applications can then be used for uncertainty propagation, sensitivity analysis, optimization, or inverse problems, as well as for correcting or improving the model predictions.

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Uzair Mansuri, PMP, CSM, PSM

Program Chief Engineer at Collins Aerospace
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I've used Machine learning (ML) to improve uncertainty quantification (UQ) in CFD models, such as turbulence modeling. An example is using a Random Forest (RF) regressor to assess uncertainty. First, collect and preprocess data from turbulence simulations. Divide the dataset into training (80%) and test (20%) sets. Train the RF model on the training set, optimizing hyperparameters using cross-validation. Evaluate the model's predictive performance on the test set using metrics like R-squared, RMSE, and MAE. Uncertainty is quantified through the variance in RF predictions, calculated as the variance of individual decision tree predictions within the forest. Such ML-based UQ approach enhances understanding of uncertainties in CFD models.

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Andrew Fiore, PhD

Chemical Engineer | Modeling and Simulation
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Before applying ML/UQ methods to CFD, it's first important to define an appropriate response variable for the given engineering application. The choice is not always obvious, but is usually some integral of a CFD field, e.g. impeller power input or average kLa, or another scalar quantity, e.g. mixing time. With a suitable response variable defined, ML approaches can be used to approximate the discrepancy between models of different fidelity and between models and experiments. Neural networks are one choice to do this given enough data. Other approaches, such as the Kennedy-O'Hagan model, use Bayesian methods instead.

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Raj Saini, PhD

Founder & Technology Specialist at R.K.S. Technology & Services? | Customized Solutions & Services | Energy Technology | R&D | CFD: AI-ML integrations | Alma Mater-IITB & IITM
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Key Metrics for Uncertainty Quantification in CFD Models: 1. Mean Squared Error (MSE) - Measures average prediction error. 2. Predictive Interval Coverage Probability (PICP) - Assesses the proportion of true outcomes within the predicted interval. 3. Width of the Prediction Interval (WPI) - Evaluates the uncertainty range. These metrics together provide a thorough assessment of model reliability, ensuring robust CFD predictions.

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Raj Saini, PhD

Founder & Technology Specialist at R.K.S. Technology & Services? | Customized Solutions & Services | Energy Technology | R&D | CFD: AI-ML integrations | Alma Mater-IITB & IITM
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Best Methods for Quantifying Uncertainty in CFD Models Using Machine Learning: 1. Bayesian Neural Networks (BNNs) - Provides a probabilistic approach & Yields a distribution of predictions. 2. Monte Carlo Dropout - Randomly drops neurons during training; Approximates Bayesian methods & Captures model uncertainty effectively. These methods help make CFD simulations more reliable by accounting for prediction variability.

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2 Methods for uncertainty estimation

When estimating uncertainty in CFD models using ML, the type and availability of data, model complexity, and desired level of accuracy and interpretability must be taken into account. Monte Carlo methods involve repeated sampling or perturbation of the input data or model parameters to generate a multitude of output scenarios. Bayesian methods use probabilistic models and inference techniques to represent the uncertainty in the input data, model parameters, and output predictions, updating them based on new evidence. Deep learning methods deploy neural networks and advanced architectures such as variational autoencoders, generative adversarial networks, or Bayesian neural networks to learn complex nonlinear mappings between inputs and outputs, capturing the uncertainty in the model weights or outputs.

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Andrew Fiore, PhD

Chemical Engineer | Modeling and Simulation
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Bayesian methods using Monte Carlo sampling of the posterior distribution are powerful uncertainty and parameter estimators that can combine data from low fidelity models, high fidelity models, and experiments within one framework. However, the computational cost of these methods is often quite high when applied to CFD models because drawing each random sample requires solving the CFD model again. Reduced-order/surrogate model approximations that can be rapidly evaluated, e.g. response surfaces or Gaussian process models, are therefore essential to applying these methods in practice.

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3 Metrics for uncertainty evaluation

When evaluating the quality and performance of uncertainty estimation methods in CFD models using ML, there are various metrics to consider, depending on the purpose and context of the analysis. Calibration measures how well the estimated uncertainty reflects the true uncertainty or error in the model predictions, while sharpness gauges how narrow or precise the estimated uncertainty is. Coverage, on the other hand, evaluates how often the estimated uncertainty contains the true value or outcome of the model predictions. A well-calibrated, sharp, and high-coverage method should produce uncertainty estimates that match the observed frequency or probability of outcomes, are as small as possible without sacrificing accuracy, and cover a large proportion of true values or outcomes without being too wide or vague.

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Paul M Coyle

Fire Engineering and Fire Risk Management - Bristol
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Parametric modelling maybe able to bridge some challenges in code - using grasshopper and Rhino for the actual structure under analysis.

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4 Here’s what else to consider

This is a space to share examples, stories, or insights that don’t fit into any of the previous sections. What else would you like to add?

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What are the best methods and metrics for quantifying uncertainty in CFD models using machine learning?

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1 Machine learning for uncertainty quantification

2 Methods for uncertainty estimation

3 Metrics for uncertainty evaluation

4 Here’s what else to consider

Computational Fluid Dynamics (CFD)

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