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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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30 Samples
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100 Samples
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300 Samples
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1000 Samples
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10000 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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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