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 given an 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