jesper kristensen (joint work with prof. n. zabaras*) · uncertainty quantification with surrogate...
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UNCERTAINTY QUANTIFICATION WITH SURROGATE MODELS IN
ALLOY MODELING Jesper Kristensen,
(joint work with Prof. N. Zabaras*)
Applied and Engineering Physics & Materials Process Design & Control Laboratory
Cornell University 271 Clark Hall, Ithaca, NY 14853-3501
and *Warwick Centre for Predictive Modelling
University of Warwick, Coventry, CV4 7AL, UK
Warwick Centre for Predictive Modelling Seminar Series, University of Warwick, Coventry, UK, January 8, 2015
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FINDING THE BEST MATERIALS
q Single candidate property (one column to the left) Ø Hours
q Million+ candidates
Ø Infeasible!
2
Search for target material/property
Unmodified image in: G. Ceder and K. Persson. Scientific American (2013)
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FINDING THE BEST MATERIALS
q Single candidate property (one column to the left) Ø Hours
q Million+ candidates
Ø Infeasible!
q Need for surrogate models Ø Rapid configuration space
exploration Ø Allows design of materials
3
Search for target material/property
Unmodified image in: G. Ceder and K. Persson. Scientific American (2013)
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USING SURROGATES IN ALLOY MODELING
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THE CLUSTER EXPANSION q Alloy surrogate model
Ø The cluster expansion
q Cluster with n points: n-pt cluster
q Expansion coefficients Jk: ECI Ø Effective cluster
interactions
q Clusters similar under space group symmetries Ø Same ECI Ø “High symmetry:
few unknowns”
5
Cluster Expansions: J. Sanchez, F. Ducastelle, and D. Gratias, Physica A (1984)
basis functions
basis functions = clusters
E(�) ⇡X
i
Ji�i +X
i,j
Ji,j�i�j + · · ·+ (M)
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CAN INFORMATION THEORY IMPROVE THERMODYNAMIC ALLOY MODELING WITH
SURROGATES?
J. Kristensen, I. Bilionis, and N. Zabaras. Physical Review B 87.17 (2013)
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COMMON METHODOLOGY q Expensive data set
q Much-used approach: Least squares
7
ECI Design matrix
Can we do better if the objective is to obtain the ground states?
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FITTING THE BOLTZMANN DISTRIBUTION
q Partition function
8
Replace Boltzmann with surrogate distribution
Ab initio energy: expensive σ: Configurational states of the system
ECI
We aim to match distributions rather than energies!
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QUANTIFY INFORMATION LOSS
9
“Distance” of choice:
True Model
S[�] =
Z
Mp(�) ln
✓p(�)
p(�|�)
◆d� � 0
Choose ECI to minimize area!
Relative Entropy (Rel Ent)
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REL ENT VS. LEAST SQUARES
10
We ideally minimize
Least squares ideally minimizes L[�] =
X
�
(E(�)� E(�;�))2
Gaussian approximation of S[�] =
Z
Mp(�) ln
✓p(�)
p(�|�)
◆d�
Matching distributions becomes a weighted least squares problem (from minimizing S above) with weights
(IN � pN1N )diag(pN )(IN � pN1N )t
pN :=
✓exp(��E(�(1)
))
ZN, · · · , exp(��E(�(N)
))
ZN
◆(t = transpose)
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Rel Ent Behavior
11
Relative Entropy
Least Squares
Energy
Low T
Sta
tes
Ignore high energy states
Relative Entropy
Least Squares
High T
All states equally likely
Energy
Sta
tes
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HOW TO COMPUTE PHASE TRANSITIONS
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Phase SpaceProbability
Resample
THERMODYNAMICS USING MCMC
13
Particle approximation
More steps taken è
Thermodynamic quantity Q (specific heat in case of phase transition)
(step s)
Fully parallelizable: Each particle on its own core
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ASMC ALGORITHM: IMPLEMENTATION
14
Adaptive step size according to how much distribution changes
Threshold: Re-locate particles
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CASE STUDY: SILICON GERMANIUM
Predict two-phase coexistence to disorder phase transition at 50 % composition
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50 % SI-GE: FIT TO ENERGIES q Use ASMC to obtain phase transitions
16
~325 K
A. van de Walle and G. Ceder, J. Phase Equilibria 23, 348 (2002)
Method: Least Squares (ATAT)
+traditional MCMC
Least squares ≈ relative entropy
~325 K
two-phase coexistence
disorder
Existing literature
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SILICON GERMANIUM q Why the similarity?
17
Small difference (< 0.04 meV)
Region of transitions
Note: This is not probabilistic, but gives an idea of the behavior versus temperature
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CASE STUDY: MAGNESIUM LITHIUM
Predict order/disorder phase transitions at 33 %, 50 %, and 66 % Mg
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MAGNESIUM LITHIUM
19
R. Taylor, S. Curtarolo, and G. Hart, Phys. Rev. B (2010)
33 % Mg composition: ~190 K 50 % Mg composition: ~300-450 K 66 % Mg composition: ~210 K
C. Barrett and O. Trautz. Trans. Am. Inst. 175 (1948)
All compositions: ~140-200 K
Fitting observed energies (cluster expansion)
Experimental (non-conclusive)
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20 R. Taylor et al. Phys. Rev. B. 81 (2010)
Method: Genetic Algorithm+MCMC
33 % Mg
50 % Mg
66 % Mg
Experiments: 140-200 K
Experiments: 140-200 K
Experiments: 140-200 K (largest error)
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MAGNESIUM LITHIUM q Why the difference?
21
Large Difference (~0.5 meV)
Region of transitions
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RELATIVE ENTROPY CONCLUSIONS
22
Alloy x Fit energies (our work)
Fit energies (literature)
Relative Entropy
Experiment
SixGe1-x 50 % ~339 K ~325 K ~339 K N/A
MgxLi1-x 33 % ~226 K ~190 K ~170 K ~140-200 K
50 % ~304 K ~300-450 K ~214 K ~140-200 K
66 % ~207 K ~210 K ~240 K ~140-200 K (largest error)
q Summary table
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BAYESIAN APPROACH TO PREDICTING MATERIALS
PROPERTIES
PROPAGATING UNCERTAINTY FROM A SURROGATE TO, E.G., A PHASE TRANSITION
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Fully Bayesian Approach q The former part of this presentation does not per se offer ways of
answering central questions such as:
Ø What is the uncertainty in a quantity of interest (e.g., a phase transition) given that we do not know the best cluster expansion and that we have limited data?
q When computing a phase transition we want to know how uncertain
we are about its value
Ø Unknown whether what we predict is OK
q The Bayesian approach can provide answers to such questions q We show how surrogate models can be used to accomplish this
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Fully Bayesian Approach q Probability means a reasonable degree of belief*
q Prior belief on clusters + ECI q Likelihood function
Ø Given a model (i.e., set of clusters + ECI) how likely is D
q Posterior belief on clusters + ECI
q Use Bayes theorem** to update degree of belief upon receiving new evidence D (what D is depends on the application)
25
p(�|D) =p(D|�)p(�)
p(D)
*Laplace, Analytical Theory of Probability (1812) **Bayes, Thomas. Philosophical Transactions (1763)
Note: D is limited, we can only see so many observations
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Propagating Uncertainty
26
Prior on property (quantity of interest) “I” Prior on cluster and ECI “θ”
q Then we observe an expensive data set D which helps us to learn more about the clusters and the ECI Ø In this work the data set was expensive energy computations
Likelihood: What information the data contains about the clusters and ECI
Notice how we integrate out the clusters and the ECI! (In principle) all cluster and ECI choices (models) consistent with D are considered
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The Quantity of Interest
27
q Different quantities of interest I can require different data sets D q This framework allows for very general quantities of interest I
q Some examples:
Ø I = phase transition • The phase transition is found from the internal energy • The data set D consists of high-accuracy (expensive) energies
Ø I = ground state line • The ground state line is found from the internal energy as well • The data set D consists of expensive energies
Ø I = maximum band gap structure • The maximum band gap structure is found, e.g., from knowing the
band gap of each structure or the entire band diagram • The data set D consists of expensive band gaps
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Propagating Uncertainty In Surrogate
28
How likely is the cluster expansion upon seeing D including what we knew before? (Bayes theorem!)
Posterior on truncation and ECI
q We stress here that we now have a probability distribution on the
property—not a single estimate Ø From this, the uncertainty estimate follows
q We now have a way to propagate uncertainty from the cluster expansion to the quantity of interest
q Next, we select a Bayesian posterior
Posterior on property
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Bayesian Posterior using a Surrogate
29
q Choose Bayesian posterior*
q Based on LASSO-inspired priors
Ø Models describing physics are typically sparse**
q Expectation values with this posterior not in closed form! Ø Resolution: MCMC Sampling
*C. Xiaohui, J. Wang, and M. McKeown (2011)
p(✓|D) /�(k)B(k, p� k + 1)||J ||�k1 ||y �XJ ||�n
2
Penalize size of ECI
k = model complexity (# of clusters)
Clusters
ECI
Size of D
p = arbitrary max set of clusters to be used LASSO Regularization
**L. Nelson et al. Physical Review B 87.3 (2013)
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Motivating Model Selection
q There is an infinite number of cluster expansions (each symbolized
by its own θ in the integral above)
Ø Which ones are most relevant to determining the value of the integral?
q We now explore model selection as an option
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CAN MODEL SELECTION BE USED TO QUANTIFY
EPISTEMIC UNCERTAINTIES WITH LIMITED DATA?
J. Kristensen and N. Zabaras. Computer Physics Communications 185.11 (2014)
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MODEL SELECTION SELECTION OF BOTH BASIS FUNCTIONS AND
EXPANSION COEFFICIENTS
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Reversible Jump Markov Chain Monte Carlo
33
q We used reversible jump Markov chain Monte Carlo (RJMCMC)* to perform the model selection
q Define 3 move types:
q Practically speaking it behaves like a standard MCMC chain Ø Use 50 % burn-in Ø Use thinning if you want to
(for memory reasons, e.g.)
Birth step (+1) Death step (-1) Update step (0)
(Birth step)
*P. Green. Biometrika 82.4 (1995)
add cluster?
Basis set (clusters) represented as a binary string (ECI not shown)
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RJMCMC CHAIN: ALGORITHM
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Model selection of clusters
Initialization
*N. Metropolis, et al. The journal of chemical physics 21.6 (1953)
Standard Metropolis-Hastings*
Model selection of the ECI
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RESULTS ON REAL ALLOYS
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MODEL SELECTION RESULTS
36
Sparse solution!
Any particular blue point in upper plot represents a cluster expansion truncation: 1) y-axis measures number of included clusters (but not which). 2) Actual values of ECI not shown
Noise agrees with DFT
p(✓|D) /�(k)B(k, p� k + 1)||J ||�k1 ||y �XJ ||�n
2
Million steps (after 50 % burn-in)
+ RJMCMC
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GROUND STATE LINE UNCERTAINTY
37
q Bayesian uncertainty in ground state line with limited data
q This is the uncertainty induced in the quantity of interest from the uncertainty in the surrogate model Ø If error bars too large: you need to increase/change your data set!
Predictive variance of ground state line is around 12 %
% Mg in MgLi
p(I|D, ·) =Z
d✓�(I[f(·; ✓)]� I)p(✓|D, ·) I = ground state line
Data set from VASP* (expensive energies)
*G. Kresse and J. Hafner. Physical Review B 47.1 (1993)
Material: MgLi
We can implicitly conclude whether the data set is large enough!
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PHASE TRANSITION UNCERTAINTY
38
q Bayesian uncertainty in phase transition from two-phase coexistence to disorder with limited data
q This is the uncertainty induced in the quantity of interest from
the uncertainty in the surrogate model Ø If error bars too large: you need to increase/change your data set!
Predictive variance of phase transition is around 6 %
19 K
p(I|D, ·) =Z
d✓�(I[f(·; ✓)]� I)p(✓|D, ·)I = two-phase coexistence to disorder phase transition
Material: SiGe at 50 %
Transitions computed via ASMC
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USING SURROGATES FOR DESIGNING MATERIALS
J. Kristensen and N. Zabaras. In review (2014)
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MATERIALS BY DESIGN
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q We are now confident about the predictive capabilities of surrogate models
q Can we also use surrogates for designing new structures with specified properties?
Ø Materials by design
q Application: Ø Optimize thermal conductivity in nanowires*
• Heat dissipation in nanochips • Thermoelectric materials
– Solar cells – Refrigeration
Ø But: Nanowires require a different way of using the cluster expansion • by “the cluster expansion” we mean the standard bulk expansion
implemented in, e.g., ATAT** • We show shortly how we addressed this issue
**A. Walle, M. Asta, and G. Ceder. Calphad 26.4 (2002) *N. Mingo et al. Nano Letters 3.12 (2003)
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WHICH SI-GE NANOWIRE CONFIGURATION MINIMIZES THE
THERMAL CONDUCTIVITY?
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DESIGN GOAL
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q Find the configuration with lowest thermal conductivity
Green-Kubo method:*
using microscopic heat current:
and a Tersoff** potential energy b/w bonds:
*R. Kubo Journal of the Physical Society of Japan 12.6 (1957) *M. Green The Journal of Chemical Physics 20.8 (1952)
**J. Tersoff Physical Review B 39.8 (1989)
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NANOWIRE CHALLENGE: LOW SYMMETRY
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q Alloy optimization problem
Ø Use cluster expansion surrogate
q Problem for nanowires: Ø Low-symmetry system Ø ECI become layer-dependent close
to surfaces Ø Easily thousands of unknowns!
q Energy is additive so we can write*:
�HCE
f = �Hvol
f +�Hsurf
f
*D. Lerch et al. Modelling and Simulation in Materials Science and Engineering 17.5 (2009)
Bulk part Surface part
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NEW CLUSTER EXPANSION APPROACH
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q Idea: embed structure of any geometry in sea of ghost sites q Group clusters under bulk symmetries
Ø We get bulk contribution alone, but for any geometry!
Silicon
Germanium Ghost site
3-pt cluster(a) (b)
Green atoms: Atoms part of 3-pt cluster (visual aid)
�HCE
f = �Hvol
f +�Hsurf
f
Could be any shape on any lattice: bcc sphere, fcc nanowire, sc 2D sheets, etc.
Ghost lattice method (GLM)
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GLM ON NANOWIRE PROJECT
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q Nanowire implementation with the GLM Ø Two different representations of the same wire (OVITO* used for
visualization)
ATAT representation
ghost site
SiGe
one periodic imageof the wire
“image” Si
“image” Ge
End view Side view
(a) (b)
1.88 nm
LAMMPS representation
GeSi
1.5 nm x
Compute correlation functions with ATAT modified for GLM (i.e., modified to parse ghosts) Compute thermal conductivity in LAMMPS
*A. Stukowski. Modelling and Simulation in Materials Science and Engineering 18.1 (2010)
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VERIFY GENERAL LAMMPS IMPLEMENTATION
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q Bulk Si and Ge (easy case): Use method in Ref. [*] in LAMMPS**
We predict 170 W/m.K for Silicon. Experimental value = 150 W/m.K. We predict 90 W/m.K for Germanium. Experimental value is 60 W/m.K. Tersoff is known to overshoot. We obtain great agreement!
*J. Chen, G. Zhang, and B. Li. Physics Letters A 374.23 (2010) **S. Plimpton. Journal of computational physics 117.1 (1995)
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VERIFY NANOWIRE IMPLEMENTATION
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q Compare data with Ceder’s group at MIT*
W/mK Pure Si wire PPG (defined later)
Our work (LAMMPS)
4.1 +/- 0.4 0.12 +/- 0.03
Ceder group (XMD)
4.1 +/- 0.3 0.23 +/- 0.05
Main sources of discrepancy • thermalization techniques • MD software • thermalization times
Annealed heating of the (relatively large) surface area was necessary.
κ
Convergence … when?
*M. Chan et al. Physical Review B 81.17 (2010)
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Nanowire Data set q 140 wires each with random
Si/Ge configuration Ø This is the random nanowire
dataset (RW) q Split RW into train and test
sets Ø Train CE with GLM on train
q Additional data sets:
Ø Planes of pure Ge (PPG) Ø Similar to PPG (SPPG)
• Perturbed: atom(s) from plane swapped with atom (s) from non-plane region
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(a)
(b)
15
10
5
0.2 0.4 0.6 0.8 1.0
Num
ber
of nan
owir
es20
1.20
1.4
RW SPPG
PPG
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CE-GLM Fit on Random Nanowires
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CE
-GLM
(W
/m.K
)
Molecular dynamics (W/m.K)
CE
-GLM
(W
/m.K
)
0 0.2 0.4 0.6 0.8 1.0 1.2 1.4
(a)
(b)
0
0.2
0.4
0.6
0.8
1.0
1.2
1.4
1.6
1.6
0
0.2
0.4
0.6
0.8
1.0
1.2
1.4
1.6
RW trainRW testSPPGPPG
Using the new cluster expansion surrogate approach to fit the nanowire data set
Now that we have surrogate; find global minimum
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Lowest Thermal-Conductivity Structure? q We find the PPG to have
lowest thermal conductivity
q Very strong case for the GLM
Ø Evidence that thermal conductivity of nanowires is well captured by first term
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CE
-GLM
(W
/m.K
)
Molecular dynamics (W/m.K)
CE
-GLM
(W
/m.K
)
0 0.2 0.4 0.6 0.8 1.0 1.2 1.4
(a)
(b)
0
0.2
0.4
0.6
0.8
1.0
1.2
1.4
1.6
1.6
0
0.2
0.4
0.6
0.8
1.0
1.2
1.4
1.6
RW trainRW testSPPGPPG
SPPGs generally lower than RW train and test sets as expected
�HCE
f = �Hvol
f +�Hsurf
f
(in our case)
≈ 0
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Compare with Literature
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From Ref. [*] on the same problem (but using a slightly different surrogate model) They found as well that the PPG wire has lowest κ
*M. Chan et al. Physical Review B 81.17 (2010)
(this image of the PPG wire is from Ref. [*])
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CONCLUDING REMARKS
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Work in Progress q Quantify uncertainties in
Ø Band gaps Ø Energies Ø Phase Diagrams Ø Thermal Conductivities Ø Any material property
q Use information theory to design materials
q Help improve how data is collected (and the resources spent in doing so) in general
Ø Choosing the limited data set in most informed way
q What happens to uncertainty quantification across length and time scales?
Ø How do uncertainties in microscopic properties affect macroscopic properties?
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