Bayesian Multi-Fidelity Optimization under Uncertainty Phaedon S. - - PowerPoint PPT Presentation

Bayesian Multi-Fidelity Optimization under Uncertainty

Maximilian Koschade maximilian.koschade@tum.de Phaedon S. Koutsourelakis p.s.koutsourelakis@tum.de

Continuum Mechanics Group Department of Mechanical Engineering Technical University of Munich, Germany

SIAM Computational Science and Engineering, Atlanta, 2017

1

Optimization under Uncertainty

Example: Material property as random field λ (x)

• z

: design variables (topology or shape)

• θ ∼ pθ (θ) : stochastic influences, e.g.
• material : discretized random field λ (x)
• temperature / load : stochastic process
• manufacturing tolerances : distributed around nominal value

Figure 1: Cross section view of stiffening rib

Introducing uncertainty to optimization problems In many engineering applications deterministic optimization is a simplification neglecting aleatory and epistemic uncertainty.

2

Optimization under Uncertainty - Objective Function

Maximize the expected utility z∗ = arg max

z

V (z) = arg max

z

• U (z, θ) pθ (θ) dθ
• z

: design variables

• θ ∼ pθ (θ) : stochastic influences on the system

Example: minimize probability of failure U (z, θ) = 1A (z, θ) (where A the event of non-failure) Example: design goal utarget U (z, θ) = exp

• − 1

2τQ (u (z, θ) − utarget)2

τQ : penalty parameter enforcing the design goal

3

Probabilistic Formulation of Optimization under Uncertainty

Reformulation as Probabilistic Inference1 Solution is given by an auxiliary posterior distribution2 π (z, θ) V (z) ∝

• π (z, θ)

posterior

dθ ∝

• U (z, θ)

likelihood

pθ (θ)

prior

dθ since the marginal π (z) ∝ V (z), given a flat prior pz (z). Conducive to consistent incorporation of epistemic uncertainty due to approximate, lower-fidelity solvers!

1Mueller (2005) 2This approach should NOT be confused with Bayesian optimization

4

Probabilistic Formulation of Optimization under Uncertainty

Reformulation as Probabilistic Inference1 Solution is given by an auxiliary posterior distribution2 π (z, θ) V (z) ∝

• π (z, θ)

posterior

dθ ∝

• U (z, θ)

likelihood

pθ (θ)

prior

dθ

• pz (z)

flat prior

since the marginal π (z) ∝ V (z), given a flat prior pz (z). Conducive to consistent incorporation of epistemic uncertainty due to approximate, lower-fidelity solvers!

1Mueller (2005) 2This approach should NOT be confused with Bayesian optimization

4

Example: Stochastic Poisson Equation

x2 x1 z (x2) ∇ · (−λ (x) ∇u (x)) = 0 dim (z) = 21 dim (θ) = 800 Solution Target x2 u (x2) u (x2) z : Control heat influx θ : Log-Normal conductivity field θ(2) θ(1) x2

5

Solution via rank-1-perturbed Gaussian q∗

−200 −150 −100 −50 50 100 150 200 0.2 0.4 0.6 0.8 1 heat fmux g(x2) x2 z∗ Sensitivity

Figure 2: Black-box stochastic variational inference in dimension 821 (dim (θ) = 800,dim (z) = 21) (Hoffman et al., 2013; Ranganath et al., 2013)

6

Solution via rank-1-perturbed Gaussian q∗

−200 −150 −100 −50 50 100 150 200 0.2 0.4 0.6 0.8 1 heat fmux g(x2) x2 z∗ Sensitivity

Cost : O

• 103

forward evaluations

6

• high dimension
• expensive numerical model

⇒ probabilistic inference can quickly become prohibitive. How can we address this issue?

6

Introduction of approximate solvers

If we denote a = log U and y = [z, θ]T we can rewrite π (y) πa (y) ∝ U (y) py (y) = exp (a (y)) py (y) =

• exp (a) δ (a − log U (y)) py (y) da

=

• exp (a) p (a|y) py (y) da

Approximate solvers = Epistemic uncertainty

• As long as p (a|y) is a Dirac, we recover posterior perfectly
• Introduction of cheap, approximate solvers leads to dispersion
• f p (a|y) and irrevocable loss of information regarding y
• We can consistently incorporate this epistemic

uncertainty in the Bayesian framework

7

Introduction of approximate solvers

If we denote a = log U and y = [z, θ]T we can rewrite π (y) πa (y) ∝ U (y) py (y) = exp (a (y)) py (y) =

• exp (a) δ (a − log U (y)) py (y) da

=

• exp (a) p (a|y) py (y) da

Regression Model We may learn p (a|y) from e.g. a Bayesian regression model or a Gaussian process GP a = φ (y)T w + ǫ This approach is impractical for a high-dimensional probability space y = [z, θ]T !

7

Introduction of approximate solvers

If we denote a = log U and y = [z, θ]T we can rewrite π (y) πa (y) ∝ U (y) py (y) = exp (a (y)) py (y) =

• exp (a) δ (a − log U (y)) py (y) da

=

• exp (a) p (a|y) py (y) da

Suppose instead we introduce a low-fidelity log-likelihood A p (a|y) =

• p (a, A|y) dA =
• p (a|A, y) p (A|y) dA

≈

• p (a|A) δ (A − log ULowFi) dA := pA (a|y)

⇒ πA (y) ∝

• exp (a) pA (a|y) py (y) da

7

Learning p (a|y) : Probabilistic multi-fidelity approach3

Introduce low-fidelity log-likelihood A pA (a|y) ≈

• p (a|A) δ (A − log ULowFi. (y)) dA
• 106
• 104
• 102
• 100
• 106
• 104
• 102
• 100

Low-Fidelity A High-Fidelity a

• Pred. density pA (a|A)
• belief of high-fidelity a

given low-fidelity A

• learn from a limited set
• f forward solver

evaluations D

• D = {a (yn) , A (yn)}N

n=1 8

Learning p (a|y) : Probabilistic multi-fidelity approach3

Introduce low-fidelity log-likelihood A pA (a|y) ≈

• p (a|A) δ (A − log ULowFi. (y)) dA
• 106
• 104
• 102
• 100
• 106
• 104
• 102
• 100

Low-Fidelity A High-Fidelity a

• Pred. density pA (a|A, D)
• belief of high-fidelity a

given low-fidelity A

• learn from a limited set
• f forward solver

evaluations D

• D = {a (yn) , A (yn)}N

n=1 8

Learning p (a|y) : Probabilistic multi-fidelity approach3

Introduce low-fidelity log-likelihood A pA (a|y) ≈

• p (a|A) δ (A − log ULowFi. (y)) dA
• 106
• 104
• 102
• 100
• 106
• 104
• 102
• 100

Low-Fidelity A High-Fidelity a Learn pA (a|A, D)

• Learn predictive density
• Using e.g. variational

relevance vector machine (VRVM) or Variational Heteroscedastic Gaussian Process (VHGP)

• D = {a (yn) , A (yn)}N

n=1 8

Learning p (a|y) : Probabilistic multi-fidelity approach3

Introduce low-fidelity log-likelihood A pA (a|y, D) ≈

• p (a|A, D) δ (A − log ULowFi. (y)) dA
• 106
• 104
• 102
• 100
• 106
• 104
• 102
• 100

Low-Fidelity A High-Fidelity a

• Pred. density pA (a|A, D)
• belief of high-fidelity a

given low-fidelity A

• learn from a limited set
• f forward solver

evaluations D

• D = {a (yn) , A (yn)}N

n=1 8

Extended Probability Space - Illustration

δ (a − log U (y)) πa(y) 1 y a πA (a, y)

9

Extended Probability Space - Illustration

πa(y) πA(y) pA (a|y) 1 2 Increase of epistemic uncertainty y a πA (a, y)

9

Extended Probability Space - Illustration

πa(y) πA(y) pA (a|y) 1 2 3 Increase of epistemic uncertainty y a πA (a, y)

9

Multi-Fidelity posterior πA (y)

Approximate πA (y) If predictive density p (a|y) is given by a Gaussian N

• a
• µ (A (y)) , σ2 (A (y))
• , then we obtain

log πA (y) = µ (A (y)) + 1 2 σ2 (A (y)) + log py (y) Place probability mass on y associated with

10

Multi-Fidelity posterior πA (y)

Approximate πA (y) If predictive density p (a|y) is given by a Gaussian N

• a
• µ (A (y)) , σ2 (A (y))
• , then we obtain

log πA (y) = µ (A (y)) A + 1 2 σ2 (A (y)) + log py (y) Place probability mass on y associated with (A): high predictive mean µ (y)

10

Multi-Fidelity posterior πA (y)

Approximate πA (y) If predictive density p (a|y) is given by a Gaussian N

• a
• µ (A (y)) , σ2 (A (y))
• , then we obtain

log πA (y) = µ (A (y)) A + 1 2 σ2 (A (y)) B + log py (y) Place probability mass on y associated with (A): high predictive mean µ (y) (B): large epistemic uncertainty σ2 (y)

10

Example: Stochastic Poisson Equation

x2 x1 z (x2) ∇ · (−λ (x) ∇u (x)) = 0 dim (z) = 1 dim (θ) = 256 Solution Target x2 u (x2) u (x2) z : Control heat influx θ : Log-Normal conductivity field θ(2) θ(1) x2

11

Effect of lower-fidelity solvers4

20 30 40 50 60 70 80 90 100 5 10 15 design variable z density estimate πA (z|D) z∗|Reference (32x32) πA (z|D) (8x8)

Figure 2: dim (θ) = 256, speedup S4×4 ≈ 2, 000 , N = 200 training data samples, density estimate obtained using MALA

4here the low-fidelity solvers are simply coarser discretizations of the stochastic

Poisson equation

12

Effect of lower-fidelity solvers4

20 30 40 50 60 70 80 90 100 5 10 15 design variable z density estimate πA (z|D) z∗|Reference (32x32) πA (z|D) (8x8) πA (z|D) (4x4)

Figure 2: dim (θ) = 256, speedup S4×4 ≈ 2, 000 , N = 200 training data samples, density estimate obtained using MALA

4here the low-fidelity solvers are simply coarser discretizations of the stochastic

Poisson equation

12

Effect of training data D

35 40 45 50 55 60 65 5 10 15 20 design variable z density estimate πA (z|D) z∗|Reference 10 data points

Figure 3: The data restricted likelihood becomes more confident due to the reduction of epistemic uncertainty by additional training samples. (dim (θ) = 256, averaged for 100 random sub-samplings of the data)

.

13

Effect of training data D

35 40 45 50 55 60 65 5 10 15 20 design variable z density estimate πA (z|D) z∗|Reference 10 data points 50 data points

Figure 3: The data restricted likelihood becomes more confident due to the reduction of epistemic uncertainty by additional training samples. (dim (θ) = 256, averaged for 100 random sub-samplings of the data)

.

13

Effect of training data D

35 40 45 50 55 60 65 5 10 15 20 design variable z density estimate πA (z|D) z∗|Reference 10 data points 50 data points 500 data points

Figure 3: The data restricted likelihood becomes more confident due to the reduction of epistemic uncertainty by additional training samples. (dim (θ) = 256, averaged for 100 random sub-samplings of the data)

.

13

Review

Summary:

• Optimization under uncertainty can be reformulated as

Bayesian inference

• Allows consistent incorporation of epistemic uncertainty

introduced by cheaper, approximate (probabilistic) models

• Inevitable loss of information, but me way obtain multi-fidelity

posterior which contains the optimal design z∗ (MAP)

• Approach applicable to any problem of Bayesian inference

Outlook:

• introduction of multiple predictors A(p)
• adaptive enrichment of training data D
• more flexible approach to learn p (a|A)

14

Review

Summary:

• Optimization under uncertainty can be reformulated as

Bayesian inference

• Allows consistent incorporation of epistemic uncertainty

introduced by cheaper, approximate (probabilistic) models

• Inevitable loss of information, but me way obtain multi-fidelity

posterior which contains the optimal design z∗ (MAP)

• Approach applicable to any problem of Bayesian inference

Outlook:

• introduction of multiple predictors A(p)
• adaptive enrichment of training data D
• more flexible approach to learn p (a|A)

14

Review

Summary:

• Optimization under uncertainty can be reformulated as

Bayesian inference

• Allows consistent incorporation of epistemic uncertainty

introduced by cheaper, approximate (probabilistic) models

• Inevitable loss of information, but me way obtain multi-fidelity

posterior which contains the optimal design z∗ (MAP)

• Approach applicable to any problem of Bayesian inference

Outlook:

• introduction of multiple predictors A(p)
• adaptive enrichment of training data D
• more flexible approach to learn p (a|A)

14

Review

Summary:

• Optimization under uncertainty can be reformulated as

Bayesian inference

• Allows consistent incorporation of epistemic uncertainty

introduced by cheaper, approximate (probabilistic) models

• Inevitable loss of information, but me way obtain multi-fidelity

posterior which contains the optimal design z∗ (MAP)

• Approach applicable to any problem of Bayesian inference

Outlook:

• introduction of multiple predictors A(p)
• adaptive enrichment of training data D
• more flexible approach to learn p (a|A)

14

References

Hoffman, M. D., Blei, D. M., Wang, C., and Paisley, J. (2013). Stochastic variational

• inference. J. Mach. Learn. Res., 14(1):1303–1347.

Mueller, P. (2005). Simulation based optimal design. In Dey, D. and Rao, C., editors, Bayesian Thinking Modeling and Computation, volume 25 of Handbook of Statistics, pages 509 – 518. Elsevier. Ng, L. W. and Willcox, K. E. (2014). Multifidelity approaches for optimization under

• uncertainty. International Journal for Numerical Methods in Engineering,

100(10):746–772. Perdikaris, P., Venturi, D., Royset, J., and Karniadakis, G. (2015). Multi-fidelity modelling via recursive co-kriging and gaussian–markov random fields. In Proc. R.

• Soc. A, volume 471, page 20150018. The Royal Society.

Ranganath, R., Gerrish, S., and Blei, D. M. (2013). Black Box Variational Inference. Aistats, 33. 15

Addendum (1): generate Training Data D = {a (yn) , A (yn)}N

n=1

Batch: Generate D such that equal numbers of A (yn) fall in Ml =

• y
• π(l)

c

≤ πc (y) ≤ π(l+1)

c

• l = 1, ..., L
• π(l+1)

c

= const · π(l)

c

• πc : posterior defined low-fidelity

solver. Adaptive Refinement: Use π (y|D) as acquisition function, corresponding to large predictive mean and epistemic uncertainty π(1)

c

π(2)

c

π(3)

c

π(4)

c

Figure 4: Iso-probability lines of the coarse posterior πc (y)

16

Addendum (2): is it possible to exclude the optimal design z∗ (MAP)?

This approach will, if executed correctly, never put zero probability mass on the MAP z∗ or any other value deemed probable under the high fidelity posterior. Potential Errors:

• Generated D does not sufficiently encapsulate p (a|A)
• Regression model is not flexible enough to learn p (a|A, D)

correctly

• Approximation of intractable posterior πA (y|D) using e.g.

VB, MCMC or SMC.

17