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 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

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

Local Model

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

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)

Local Model

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

Example Problem

30 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

‘Truth’ Model Dataset used : Snapshots of temporal mixing layer data from Amirreza

Trajectory Random Draws FPVA Table

Initial condition

Results of ML scheme

Application to EUQ of RANS

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

Model Output

Not perfect, but way better

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

Random Draws

Random Draws

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