ASME V&V 2016 talk on probabilistic active subspaces

Gaussian processes with built-in dimensionality reduction .

Transcript

1. Probabilistic Active Subspaces: Learning High-dimensional Noisy Functions without Gradients Rohit Tripathy School of Mechanical Engineering Purdue University predictivesciencelab.org
2. The Team 2 Rohit Tripathy rtripath@purdue.edu PhD student Marcial Gonzalez gonza226@purdue.edu Assistant Professor Mechanics Ilias Bilionis ibilion@purdue.edu Assistant Professor Uncertainty Quantification predictivesciencelab.org
3. The Paper Submitted to Journal of Computational Physics. http://arxiv.org/abs/1602.04550 as “Gaussian processes with built-in dimensionality reduction: Applications in high-dimensional uncertainty propagation”
4. Motivation Input Parameters Physical model Quantities of interest We’ll think about it as a mathematical function: 4
5. Uncertainty Propagation Problem 5 • Given the map: • PDF: • Mean: • Variance:
6. MONTE CARLO • Simplest way to propagate uncertainty. • Convergence rate independent of number of dimensions. • Realistic problems require tens of thousands of simulations. • “Monte Carlo is fundamentally unsound” - O’ Hagen (1987). 6
7. The Surrogate Idea • Do a finite number of simulations. • Replace model with an approximation: • The surrogate is usually cheap to evaluate. • Solve the UQ problem with the surrogate. 7
8. Examples of Surrogates • generalized linear models. • generalized polynomial chaos. • Neural networks. • support vector machines. • Gaussian Processes. 8
9. Problems • The response is very expensive. • The input is high-dimensional (> 20). • We cannot compute derivatives of the response. • The response may be noisy. “How do you build a surrogate under these conditions?”
10. Curse of Dimensionality
11. Gaussian process regression 11 Prior GP Posterior GP Bayes rule
12. Active Subspaces Input Parameters Physical model Quantities of interest Reduced dim space orthogonal projection matrix Gaussian process
13. Classic approach Step 1: Find W Step 2: Regression Gradient info
14. Probabilistic Approach to Active Subspaces Measurement noise Just another parameter
15. Probabilistic Approach to Active Subspaces maximize log-likelihood Optimization over Stiefel manifold
16. Probabilistic Approach to Active Subspaces Needs to remain orthogonal maximize log-likelihood Line search with efficient global optimization (Bayesian global optimization)
17. Comparison to the Classic Approach
18. Comparison to the Classic Approach Common: 200 observations plus gradients W Probabilistic: just 200 observations W (almost) identical
19. Robustness to Noise The classic approach cannot deal with noise. How robust is our approach to noise?
20. Robustness to noise Relative L2 error in the estimation of W
21. How do we find the reduced space dimension? Use the Bayesian information criterion - a very crude approximation to model evidence -
22. Does BIC find the right active dimension? True d=1, BIC becomes flat after d=1 True d=2, BIC becomes flat after d=2 True d=3, BIC becomes flat after d=3
23. Granular Crystals • Unique, highly non-linear dynamical properties. • Formation and propagation of highly localized elastic stress w • Dynamics described by a fully elastic model known as the He
24. Granular Crystals Input parameters: Radii Young moduli Gaps Striker velocity
25. Granular Crystals Force Time
26. Particle 20 Soliton Amplitude W g Predicted-vs- observed (validation set)
27. Particle 40 Soliton Time of Flight W g Predicted-vs- observed (validation set)
28. Uncertainty Propagation
29. Next Steps • Extend to non-linear dimensionality reduction • Flat architecture - infinite mixture of AS-GP models. • Deep architecture - deep GPs with AS-GP components. • Epistemic uncertainty on W (MCMC on Stiefel manifold (Giro
30. Thank you! http://arxiv.org/abs/1602.04550 as “Gaussian processes with built-in dimensionality reduction: Applications in high-dimensional uncertainty propagation”