Epistemic UQ

1. Using Machine Learning for Epistemic Uncertainty Quantification in Combustion and Turbulence Modeling

2. 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 givenan eddy-viscosity based RANS solution • Preliminary work • We will show a way it could be done, not how it should be done

3. 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

4. Local Model

5. 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

6. 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)

7. Local Model

8. 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

9. Example Problem

10. 30 Samples

11. 100 Samples

12. 300 Samples

13. 1000 Samples

14. 10000 Samples

15. 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

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

17. Trajectory Random Draws FPVA Table

18. Initial condition

19. Results of ML scheme

20. Application to EUQ of RANS

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

22. Model Output

23. Not perfect, but way better

24. 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

25. Random Draws

26. Random Draws

27. 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