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Constraint Reasoning and Kernel Clustering for Pattern Decomposition With Scaling. Ronan LeBras Computer Science Theodoros Damoulas Computer Science Ashish Sabharwal Computer Science Carla P. Gomes Computer Science John M. Gregoire Materials Science / Physics - PowerPoint PPT Presentation

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2x2 chess board, 2 pieces, 2 steps

Constraint Reasoning and Kernel Clusteringfor Pattern Decomposition With ScalingRonan LeBrasComputer ScienceTheodoros DamoulasComputer ScienceAshish SabharwalComputer ScienceCarla P. GomesComputer ScienceJohn M. GregoireMaterials Science / PhysicsBruce van DoverMaterials Science / Physics

Sept 15, 2011CP11

1MotivationRonan Le Bras - CP11, Sept 15, 20112

Cornell Fuel Cell InstituteMission: develop new materials for fuel cells.

An Electrocatalyst must: Be electronically conducting Facilitate both reactionsPlatinum is the best known metal to fulfill that role, but: The reaction rate is still considered slow (causing energy loss) Platinum is fairly costly, intolerant to fuel contaminants, and has a short lifetime.Goal: Find an intermetallic compound that is a better catalyst than Pt.They are truly one of the leading group (if not the leading group) in fuel cell technology. Their mission, should they decide to accept it, is to develop new materials for fuel cell technology.

On one side (the anode), Hydrogen is flowing in, and on the other side (the cathode), Oxygen comes in. The reaction of H2 with O2 produces water, but it also releases electrons, which might be used to provide power to any sort of load.What we are interested in here, is these two electrocatalysts, which serve several functions. 1) First, they must be electronically conducting 2) Second, they should facilitate the reaction of fuel (oxydation of the hydrogen), and the reaction of oxygen (reduction of the oxygen). Typically, platinum is used for the electrocatalysts, as it is the best metal known to fulfill that role. However, with platinum, the reaction rate is still considered slow (causing energy loss), and platinum is fairly costly. Overall, this kinda prevents from the widespread use of fuel cells.2Motivation3Recipe for finding alternatives to PlatinumIn a vacuum chamber, place a silicon wafer.Add three metals.Mix until smooth, using three sputter guns.Bake for 2 hours at 650CTaRhPd(38% Ta, 45% Rh, 17% Pd)Deliberately inhomogeneous composition on Si wafer Atoms are intimately mixed

[Source: Pyrotope, Sebastien Merkel]Ronan Le Bras - CP11, Sept 15, 2011They have been using 100% platinum as electrocatalyst. But they would like to try any compound, such as 50% Platinum and 50% Fe. To try multiple combinations at once, they use what they call a combinatorial approach.Here is the recipe.Add three metals (here, Tantalum, Rhodium, and Palladium.By doing so, they are able to generate any combination of these three metals3Motivation4Identifying crystal structure using X-Ray Diffraction at CHESSXRD pattern characterizes the underlying crystal fairly wellExpensive experimentations: Bruce van Dovers research team has access to the facility one week every year.

TaRhPd(38% Ta, 45% Rh, 17% Pd)

[Source: Pyrotope, Sebastien Merkel]

Ronan Le Bras - CP11, Sept 15, 20114Motivation5TaRhPdRonan Le Bras - CP11, Sept 15, 20115Motivation6TaRhPd++Ronan Le Bras - CP11, Sept 15, 20116Motivation7TaRhPdRonan Le Bras - CP11, Sept 15, 20117Motivation8

TaRhPd

+Ronan Le Bras - CP11, Sept 15, 2011PiPj8Motivation9

TaRhPd+Ronan Le Bras - CP11, Sept 15, 2011PiPkPj9MotivationFeAlSi

INPUT:pure phaseregionFeAlSim phase regions k pure regions m-k mixed regions

XRD patterncharacterizingpure phasesMixedphaseregionOUTPUT:Additional Physical characteristics:Peaks shift by 15% within a regionPhase ConnectivityMixtures of 3 phasesSmall peaks might be discriminativePeak locations matter, more than peak intensities10Ronan Le Bras - CP11, Sept 15, 201110Motivation11

TaRhPdFigure 1: Phase regions of Ta-Rh-Pd

Figure 2: Fluorescence activity of Ta-Rh-PdRonan Le Bras - CP11, Sept 15, 2011If we could look at the actual crystal structure at any single point on this simplex, here is what we would get. The colored regions correspond to points that share the same crystal structure. Now, if we look at the fluorescence activity on this thin film (which characterizes electrocatalyst property), we see that the highest electrocatalyst activity coincides with the purple phase region. Our goal is to be able to identify this map of phase regions, knowing what they are made of, in order to understand what makes a good electrocatalyst and to be able to synthesize the corresponding material.11Outline12 Motivation Problem Definition Abstraction Hardness CP Model Kernel-based Clustering Bridging CP and Machine learning Conclusion and Future workRonan Le Bras - CP11, Sept 15, 201112Problem Abstraction: Pattern Decomposition with Scaling13Input

OutputRonan Le Bras - CP11, Sept 15, 2011a) Pi is the union of scaled basis patternsb) The number of basis patterns with a non-zero coefficient at vertex vi is at most M13Problem Abstraction: Pattern Decomposition with Scaling14Input

Outputv1v2v3v4v5Ronan Le Bras - CP11, Sept 15, 2011a) Pi is the union of scaled basis patternsb) The number of basis patterns with a non-zero coefficient at vertex vi is at most M14Problem Abstraction: Pattern Decomposition with Scaling15Input

Outputv1v2v3v4v5P1P2P3P4P5Ronan Le Bras - CP11, Sept 15, 2011a) Pi is the union of scaled basis patternsb) The number of basis patterns with a non-zero coefficient at vertex vi is at most M15Problem Abstraction: Pattern Decomposition with Scaling16Input

Outputv1v2v3v4v5P1P2P3P4P5M= K = 2, = 1.5Ronan Le Bras - CP11, Sept 15, 2011a) Pi is the union of scaled basis patternsb) The number of basis patterns with a non-zero coefficient at vertex vi is at most M16Problem Abstraction: Pattern Decomposition with Scaling17Input

Outputv1v2v3v4v5P1P2P3P4P5M= K = 2, = 1.5B2B1Ronan Le Bras - CP11, Sept 15, 2011a) Pi is the union of scaled basis patternsb) The number of basis patterns with a non-zero coefficient at vertex vi is at most M17Problem Abstraction: Pattern Decomposition with Scaling18Input

Outputv1v2v3v4v5P1P2P3P4P5M= K = 2, = 1.5B2B1s11=1.00s12=0.95s13=0.88s14=0.84s15=0s21=0.68s22=0.78s23=0.85s24=0.96s25=1.00Ronan Le Bras - CP11, Sept 15, 2011a) Pi is the union of scaled basis patternsb) The number of basis patterns with a non-zero coefficient at vertex vi is at most M18Problem Hardness19Assumptions: Each Bk appears by itself in some vi / No experimental noise The problem can be solved* in polynomial time.Assumption: No experimental noise The problem becomes NP-hard (reduction from the Set Basis problem)Assumption used in this work: Experimental noise in the form of missing elements in Piv1v2v3v4v5P1P2P3P4P5M= K = 2, = 1.5B2B1s11=1.00s12=0.95s13=0.88s14=0.84s15=0s21=0.68s22=0.78s23=0.85s24=0.96s25=1.00Ronan Le Bras - CP11, Sept 15, 2011P0=19Outline20 Motivation Problem Definition CP ModelModelExperimental Results Kernel-based Clustering Bridging CP and Machine learning Conclusion and Future workRonan Le Bras - CP11, Sept 15, 201120CP Model21Advantage: Captures physical properties and relies on peak location rather than height.Drawback: Does not scale to realistic instances; poor propagation if experimental noise.

Ronan Le Bras - CP11, Sept 15, 201121CP Model (continued)22Advantage: Captures physical properties and relies on peak location rather than height.Drawback: Does not scale to realistic instances; poor propagation if experimental noise.Advantage: Captures physical properties and relies on peak location rather than height.Drawback: Does not scale to realistic instances; poor propagation if experimental noise.

Ronan Le Bras - CP11, Sept 15, 201122CP Model Experimental Results23Advantage: Captures physical properties and relies on peak location rather than height.Drawback: Does not scale to realistic instances; poor propagation if experimental noise.For realistic instances, K = 6 and N 218... Ronan Le Bras - CP11, Sept 15, 201123Outline24 Motivation Problem Definition CP Model Kernel-based Clustering Bridging CP and Machine learning Conclusion and Future workRonan Le Bras - CP11, Sept 15, 201124Kernel-based ClusteringAdvantage: Captures physical properties and relies on peak location rather than height.Drawback: Does not scale to realistic instances; poor propagation if experimental noise.25 Set of features: X =

Similarity matrices:

[X.XT]Method: K-means on Dynamic Time-Warping kernelGoal: Select groups of samples that belong to the same phase region to feed the CP model, in order to extract the underlying phases of these sub-problems.Better approach: takeshifts into accountRed: similarBlue: dissimilar+

++

Ronan Le Bras - CP11, Sept 15, 2011We build a radial basis function kernel from minimum-cost alignment of two sequences.

The DTW kernel considers the minimum-cost alignment of two sequences (xi, xj), where c_i is the ith row of the minimum-cost alignment matrix.As the number of phase regions is a hidden parameter, we over-segment the kernel by choosing a large number of clusters.

25Bridging CP and ML26Goal: a robust, physically meaningful, scalable, automated solution method that combines:

Similarity Kernels& ClusteringMachine Learningfor a globaldata-driven view

UnderlyingPhysics

A language forConstraintsenforcinglocal detailsConstraint Programming modelRonan Le Bras - CP11, Sept 15, 201126Bridging Constraint Reasoning and Machine Learning: Overview of the Methodology27

FeAlSi

INPUT:Peakdetection

Machine Learning:Kernel methods,Dynamic Time WrappingMachine Learning:Partial ClusteringFeAlSiCP Model & Solveron sub-problems, +, only onlyFull CP Modelguided bypartial solutions

OUTPUTFix errorsin dataRonan Le Bras - CP11, Sept 15, 201127Experimental Validation28

Al-Li-Fe instancewith 6 phases:Ground truth(known)Our CP + ML hybridapproach is muchmore robustPrevious work (NMF)violates manyphysical requirements28Conclusion29Hybrid approach t

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