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Using Machine Learning for Epistemic Uncertainty Quantification in Combustion and Turbulence Modeling. 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 - PowerPoint PPT Presentation

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Using Machine Learning for Epistemic Uncertainty Quantification in Combustion and Turbulence Modeling1Epistemic UQUse machine learning to learn the error between the low fidelity model and the high fidelity modelWant to use it as a correction and an estimate of errorWorking on two aspects -- Approximate the real source term (in progress equation) given a RANS+FPVA solutionApproximate the real Reynolds stress anisotropy given an eddy-viscosity based RANS solution Preliminary workWe will show a way it could be done, not how it should be done2Basic IdeaWe 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 resultIf the error model is stochastic (and correct), draws from that model give us estimates of uncertainty. To make model fitting tractable we decouple the problemModel of local uncertainty based on flow-featuresModel of coupling of uncertainty on a macro scale

3Local Model4

Model Generation OutlineGet a training set which consists of low-fidelity solutions alongside the high-fidelity resultsChoose 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 features5ExampleIn the RANS/DNS case, we are interested in the RANS turbulence model errorsInput 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 Model7

SinkerFor 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 Problem9

30 Samples10

100 Samples11

300 Samples12

1000 Samples13

10000 Samples14

Combustion ModelingDNS finite rate chemistry dataset as high fidelity model, RANS flamelet model is low fidelity modelInput flow features are the flamelet table variables (mixture fraction, mixture fraction variance, progress variable)Output flow variable is source term in progress-variable equationUse a GP as the spatial fit15Truth Model16

Dataset used : Snapshots of temporal mixing layer data from Amirreza Trajectory Random Draws17

FPVA TableInitial condition18

Results of ML scheme19

Application to EUQ of RANS20

Input Data21Add in marker, normalized wall distance, and p/ as additional flow features, and use Sinker

Model Output22

Not perfect, but way better23

Generating ErrorbarsEach 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 quantitiesDraw two Gaussian random variables, and tweak the barycentric coordinate by that many standard deviations in x and yIf the point goes off the triangle, project it back onto the triangleGives us a family of new turbulence models24Random Draws25

Random Draws26

ConclusionsPromising early resultsBasic idea: Learn `mean and variance of error distribution of modeling terms in the space of FEATURESThere is a lot of work to be doneFeature selectionBetter uncertainty modeling (non-Gaussian)Kernel selectionNeed to develop a progressive / logical test suite to evaluate the quality of a model

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