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Using Machine Learning for Epistemic Uncertainty

Quantification in Combustion and Turbulence Modeling

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Epistemic UQ

• Use machine learning to learn the error between the low fidelity model and the high fidelity model– Want to use it as a correction and an estimate of error

• Working on two aspects -- Approximate the real source term (in progress equation)

given a RANS+FPVA solution– Approximate the real Reynolds stress anisotropy given an

eddy-viscosity based RANS solution • Preliminary work

– We will show a way it could be done, not how it should be done

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Basic Idea

• We can compare low fidelity results to high fidelity results and learn an error model– Model answers: “What is the true value given the low-

fidelity result”• If the error model is stochastic (and correct), draws

from that model give us estimates of uncertainty. • To make model fitting tractable we decouple the

problem– Model of local uncertainty based on flow-features– Model of coupling of uncertainty on a macro scale

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Local Model

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Model Generation Outline

• Get a training set which consists of low-fidelity solutions alongside the high-fidelity results

• Choose a set of features in high-fidelity to be learned ( y )

• Choose a set of features in low-fidelity which are good representations of the error ( x )

• Learn a model for the true output given the input flow features

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Example

• In the RANS/DNS case, we are interested in the RANS turbulence model errors

• Input of the model is RANS location of the barycentric map, the marker, wall distance, and (5 dimensional)

• Output of the model is DNS location in the barycentric map (2 dimensional)

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Local Model

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Sinker

• For a test location, each point in the training set is given a weight set by a kernel function

• Then, using the true result at the training points and the weights, compute a probability distribution over the true result

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Example Problem

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300 Samples

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1000 Samples

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Combustion Modeling

• DNS finite rate chemistry dataset as high fidelity model, RANS flamelet model is low fidelity model

• Input flow features are the flamelet table variables (mixture fraction, mixture fraction variance, progress variable)

• Output flow variable is source term in progress-variable equation

• Use a GP as the spatial fit

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‘Truth’ Model

Dataset used : Snapshots of temporal mixing layer data from Amirreza

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Trajectory Random Draws

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FPVA Table

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Initial condition

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Results of ML scheme

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Application to EUQ of RANS

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Input Data

• Add in marker, normalized wall distance, and p/ε as additional flow features, and use Sinker

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Model Output

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• Not perfect, but way better

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Generating Errorbars

• Each point also has a variance associated with it (which is an ellipse for now)

• We can use these uncertainties to generate error bars on macroscopic quantities

• Draw two Gaussian random variables, and tweak the barycentric coordinate by that many standard deviations in x and y

• If the point goes off the triangle, project it back onto the triangle

• Gives us a family of new turbulence models

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Random Draws

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Random Draws

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Conclusions

• Promising early results• Basic idea:

Learn `mean and variance’ of error distribution of modeling terms in the space of FEATURES

• There is a lot of work to be done– Feature selection– Better uncertainty modeling (non-Gaussian)– Kernel selection

• Need to develop a progressive / logical test suite to evaluate the quality of a model