The marvels of modern technology can largely be attrib-uted to the discovery and characterization of new mate-rials. The discovery of semiconductors laid the founda-

tion for modern electronics, while the formulation of newmolecules allows us to treat diseases previously thoughtincurable. Looking into the future, some of the largest prob-lems facing humanity now are likely to be solved by the dis-covery of new materials. In this article, we explore the tech-niques materials scientists are using and show how our novelartificial intelligence system, Phase-Mapper, allows materialsscientists to quickly solve material systems to infer theirunderlying crystal structures and has led to the discovery ofnew solar light absorbers.

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SPRING 2018 15Copyright © 2018, Association for the Advancement of Artificial Intelligence. All rights reserved. ISSN 0738-4602

Phase-Mapper: Accelerating Materials Discovery with AI

Junwen Bai, Yexiang Xue, Johan Bjorck, Ronan Le Bras, Brendan Rappazzo, Richard Bernstein, Santosh K. Suram, R. Bruce van Dover,

John M. Gregoire, Carla P. Gomes

� From the stone age to the bronze, iron, andmodern silicon ages, the discovery and charac-terization of new materials has always beeninstrumental to humanity’s development andprogress. With the current pressing need toaddress sustainability challenges and find alter-natives to fossil fuels, we look for solutions in thedevelopment of new materials that will allow forrenewable energy. To discover materials with therequired properties, materials scientists can per-form high-throughput materials discovery, whichincludes rapid synthesis and characterization viaX-ray diffraction (XRD) of thousands of materi-als. A central problem in materials discovery, thephase map identification problem, involves thedetermination of the crystal structure of materi-als from materials composition and structuralcharacterization data. This analysis is tradition-ally performed mainly by hand, which can takedays for a single material system. In this workwe present Phase-Mapper, a solution platformthat tightly integrates XRD experimentation, AIproblem solving, and human intelligence forinterpreting XRD patterns and inferring the crys-tal structures of the underlying materials. Phase-Mapper is compatible with any spectral demix-ing algorithm, including our novel solver,AgileFD, which is based on convolutive nonneg-ative matrix factorization (NMF). AgileFDallows materials scientists to rapidly interpretXRD patterns, and incorporates constraints tocapture prior knowledge about the physics of thematerials as well as human feedback. With oursystem, materials scientists have been able tointerpret previously unsolvable systems of XRDdata at the Department of Energy’s Joint Centerfor Artificial Photosynthesis, including the Nb-Mn-V oxide system, which led to the discovery ofnew solar light absorbers and is provided as anillustrative example of AI-enabled high-through-put materials discovery.

A powerful strategy employed by materials scientistis high-throughput materials discovery (Green,Takeuchi, and Hattrick-Simpers 2013), the idea beingto rapidly synthesize thousands of different materialsand quickly screen them for desirable properties.These materials are developed by depositing differentelements on a wafer in varying amounts, which isanalogous to atomic spray painting, where elementsare mixed with different proportions, rendering sim-ple mixtures as well as enabling the emergence of newmaterials, just as the mixture of primary colors resultsin both obvious mixtures and secondary colors.

In this article, we address the phase-mapping prob-lem, a central problem in high-throughput materialsdiscovery, which has critically lacked an efficientsolution method. At a fundamental level, the phase-mapping problem entails demixing data measure-ments in terms of a few simple components or crystalstructures, each describing a single material, subjectto intricate constraints on the solutions induced bythe physics of the underlying materials. A material’sphase describes a range of elemental composition andother conditions over which its properties and struc-ture, the arrangement of the constituent atoms,change little. X-ray diffraction (XRD) is a ubiquitoustechnique to characterize crystal phases, as it pro-duces a signal containing a series of peaks that serveas a fingerprint of the underlying atomic arrangementor crystal structure. Using traditional methods, mate-rials scientists can obtain and interpret 1 to 10 XRDmeasurements per day, and with the recent develop-ment of automated, synchrotron-based XRD experi-ments, the measurement throughput has been accel-erated to 103 to 105 measurements per day (Gregoireet al. 2009; 2014). The creation of a phase-mappingalgorithm that generates phase diagrams from thesedata remains an unsolved problem in materials sci-ence despite a series of advancements over the pastdecade (Hattrick-Simpers, Gregoire, and Kusne 2016).The problem is challenging, given that often the X-ray diffraction patterns correspond to a mixture ofcrystal structures, some of them not necessarily sam-pled individually, requiring an algorithm that candemix patterns while simultaneously identifying thebasis patterns. The most pertinent need is to generatea physically meaningful phase diagram (one that gen-erates materials science knowledge) for the materialsin a given library, or a collection of co-depositedmaterials on a substrate, which relies on the spectraldemixing of the 102–103 XRD patterns into a small setof basis patterns (typically less than 10).

The traditional analysis workflow relies on itera-tive manual analysis and heuristics, resulting in theanalysis of only a few systems a year. This quicklybecomes a bottleneck, as manual analysis cannotkeep up with the rate at which data are generated.Automatic analysis becomes imperative in order toanalyze the vast amount of data that are generated inhigh-throughput experiments.

The need for automatic and scalable tools providesunique opportunities to apply cutting-edge tech-niques in computer science and AI to accelerate thematerials discovery process. We developed Phase-Mapper, a comprehensive platform that tightly inte-grates XRD experimentation, AI problem solving,and human intelligence (figure 1), to address thiscomputational challenge. In this platform, withinminutes, an AI solver provides for the phase-map-ping problem physically meaningful results, whichare examined and potentially further refined bymaterials scientists interactively and in real time. Inaddition, the results of Phase-Mapper can be used tofurther inform future experimental designs. Thedemixing algorithm is a cornerstone of the Phase-Mapper platform. We have developed a novel solvercalled AgileFD, which is based on convolutive non-negative matrix factorization (cNMF). Nonnegativematrix factorization (NMF) is commonly used inapplications such as computer vision and topic mod-eling (Lee and Seung 2001), and cNMF extends thismethod to convolutive mixtures used in blind sourceseparation of audio signals and speech recognition(Smaragdis 2004; Mørup and Schmidt 2006). AgileFDfeatures a computationally efficient, gradient-basedsearch method using lightweight iterative updates ofcandidate solutions. In addition, AgileFD integratesfunctionalities beyond cNMF. The extensions ofAgileFD, described here, include incorporation ofconstraints to encode both human input, which cap-italizes on a researcher’s knowledge of a particulardata set, and prior knowledge of the problem relatedto the underlying physics of phase diagrams. This, asdemonstrated below, can be critical in obtainingphysically meaningful solutions. In developing thePhase-Mapper platform, careful attention has beengiven to delivering a rich suite of capabilities whilemaintaining solver convergence times within min-utes, which enables researchers to interact with thesolver to refine the solution.

We evaluate Phase-Mapper and several solvers thatwere proposed in recent years. In general, we observethat the solutions found by AgileFD and its variantsbetter match the ground truth. A vanilla NMFapproach performs poorly, as it fails to capture phys-ical constraints. Conversely, constraint program-ming–based approaches are able to enforce some ofthe physical constraints but scale poorly. We showempirically that AgileFD with its extensions is able tofind solutions that are close to the physical reality.

We first encountered the phase-mapping problemseven years ago as part of our computational sustain-ability (Gomes 2009) effort to address pressing prob-lems in renewable energy. Phase-Mapper is the cul-mination of our work since then, in closecollaboration with experts in materials science. Overthe course of this collaboration, we have madeimportant contributions to the formal characteriza-tion of this problem, developed several synthetic

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instance generators, and developed several algo-rithms with theoretical and practical guarantees. Wehave also continuously developed tools to shareexperimental instance data, results, and solutionvisualizations with our collaborators throughout (LeBras et al. 2011; Ermon et al. 2012; Le Bras et al. 2014;Ermon et al. 2015; Xue et al. 2015). Phase-Mapper isour most successful tool to date in this area: itremoves many of the practical barriers to the use ofprevious methods, including better scalability, run-times suitable for interactive use, and ease of access.

Phase-Mapper has been used at the Department ofEnergy’s Joint Center for Artificial Photosynthesis(JCAP) to run hundreds of phase-mapping solutionsin the JCAP materials discovery pipeline. Prior toPhase-Mapper, the difficulty of interpreting X-ray dif-fraction data limited JCAP scientists’ ability to takefull advantage of resources to conduct high-through-put experiments. Since the deployment of Phase-Mapper, thousands of X-ray diffraction patterns havebeen processed and the results are yielding discoveryof new materials for energy applications. These areexemplified by the discovery of a new family of met-al oxide light absorbers in the previously unsolvedNb-Mn-V oxide system, which is provided here as acase study and is an illustrative example of theimportance of encoding physical constraints toobtain physically meaningful phase diagram solu-tions. We believe Phase-Mapper will lead to furtherdevelopments in high-throughput materials discov-ery by providing rapid and critical insights into thephase behavior of new materials.

Phase-Mapper: AI for Materials Discovery

An experimentation pipeline for rapidly synthesiz-ing, characterizing, and identifying new materials isreferred to as high-throughput materials discovery orcombinatorial materials discovery. In this pipeline, ahandful of elements are deposited together on a two-dimensional substrate, so that different locations onthe substrate receive varying proportions of the ele-ments. This smooth variation in elemental composi-tion across the substrate gives rise to the forming ofa discrete set of materials, each of which is present inparticular regions of the substrate.

The deposition process is analogous to atomicspray paint, as mentioned earlier. Imagine red, green,and blue spray paint being simultaneously sprayedonto a surface (or wafer) with each color sourceplaced at the vertex of an equilateral triangle. Nearthese vertices, the deposited color appears simplyred, green, or blue, and throughout the area of thetriangle a continuum of the possible colors areobtained, where each color on the spectrum exists ata unique point on the wafer. In the same manner, thedeposited materials “library” contains a broad spec-trum of compositions (given the starting elements),

and the atoms in different composition regions mayarrange in a unique way to form a unique “phase”whose properties differ from other materials, evenother compositions and phases formed from thesame elements. It is the hope that one of these newmaterials will have a composition and phase thatexhibit the desired properties, and to fully under-stand the composition-phase-property relationships,the full phase map must be solved.

In the libraries being studied, the new materials aretypically crystalline, meaning that at the atomic scaleatoms are arranged in particular lattice structures,and the phase noted earlier is described by the sym-metry and composition of the lattice structure. On alarger length scale, typically 5 to 500 nm, the latticestructure may alternate between two or three differ-ent structures, constituting a mixed-phase material.Each phase and phase mixture can exhibit uniqueproperties, creating the need for materials scientiststo understand, for each material library, how to cate-gorize each material in the library (on the wafer) interm of its phase mixture

What data should materials scientists look at todetermine the crystal structures? An indirect way ofprobing the microscopic structure is through X-raydiffraction. When X-rays are directed against a crys-tal, atomic layers will reflect the light; and for specificangles determined by the spacing between atomiclayers, this reflected light will interfere constructive-ly, giving rise to a strong signal. Thus, by scanningthrough all angles and measuring the reflected light,materials scientists are able to infer the structure of

Figure 1. The Phase-Mapper Platform.

The Phase-Mapper platform integrates experimentation, AI problem solv-ing, and human feedback into a platform for high-throughput materials dis-covery.

Phase-Mapper

Solution and Interaction

X-ra

y D

iffra

ctio

n D

ata Experim

ental Design

HighThroughputExperiments

AISolver

HumanIntelligence

and Feedback

the atomic layers without any microscopic measure-ments. This process is done for different points on asingle wafer and because this process is fast whenusing synchrotron sources that provide a large flux ofX-rays, materials scientists can rapidly characterizehundreds or thousands of materials a day using X-raydiffraction. An example of the apparatus for takingthese XRD measurements can be seen in figure 2.

Phase MappingA key challenge in high-throughput materials dis-covery is to solve the phase-mapping problem, whichidentifies the characteristic XRD patterns of thematerials (or basis patterns or crystal structures of thematerials) that demix the XRD signals from the high-throughput experiments, some signals of which maybe pure in that they represent one phase, while oth-ers are a linear combination of the pure phases. Avisual description of the phase-mapping problem canbe seen in figure 3.

Mathematically, the measured XRD pattern in thej-th sample point can be characterized by a onedimensional signal Aj(q). The scattering vector mag-nitude (q) is a monotonic transformation of the dif-fraction angle, and is directly related to the spacingof atoms in a crystal. The phase-mapping problem isto find a small number of phases W1(q),…, WK(q),and their corresponding activation coefficients hij,such that the XRD patterns at each sample point canbe explained by a linear combination of phases:

(1)Aj q( ) � hijWi

i=1

K

� �ij ,q( )

The physical process of alloying complicates the lin-ear combination by introducing additional scalingfactors λij. Alloying typically can be approximated bya multiplicative scaling of the XRD pattern of aspecific phase in the q domain; we also refer to thisprocess as peak shifting. We use the term Wi(λij,q) toallow for the phases to scale slightly according toparameter λij at each sample point. In addition topeak shifting, there are a number of other constraintson the solution of the phase-mapping problem, aris-ing from the fact that the solution must describe asystem constrained by the laws of physics. Oneimportant precept is the Gibbs phase rule, which lim-its the number of phases present to at most k phasesper sample point, in a system involving k elements.Therefore, in a k-element system, no more than kcoefficients among hij for fixed j may be nonzero.Additionally, feasible spatial variation of hij by com-position, as well as the shapes that each Wi may take,are constrained by the relevant physics.

Fundamentally novel techniques are required tosolve the phase-mapping problem quickly and accu-rately. Historically, the phase-mapping problem hasbeen solved by hand, which can take days or monthsfor a single system, and has become the bottleneck ofthe entire materials discovery workflow. A number ofautomatic techniques have been developed in recentyears, which can be broadly grouped into clustering,constraint reasoning, and factor decompositionapproaches. Proposed clustering methods such ashierarchical clustering (Long et al. 2007), dynamictime warping kernel clustering (Le Bras et al. 2011),and mean shift theory (Kusne et al. 2014) producemaps of phase regions, but fail to resolve mixtures or

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Figure 2. Apparatus for Taking XRD Measurements.

To the left is an image of the X-ray diffraction apparatus being used to characterize a composition library in the high-through-put experiment. On the right is an illustration of X-ray diffraction. Incoming X-rays hit the composition sample. They then dif-fract and are detected by the XRD detector.

scatteredX-rays

XRDdetector

composition sample

XRDdetector

X-raysource

composition library

incomingX-rays

identify basis patterns, and do not necessarily pro-duce results consistent with physics. Constraint rea-soning approaches, including satisfiability modulotheory (SMT) methods (Ermon et al. 2012), can pro-vide physically meaningful results, but depend heav-ily on effective preprocessing, such as peak identifi-cation, and are computationally intensive.Approaches based on nonnegative matrix factoriza-tion (Long et al. 2009) are computationally efficient,but generally perform poorly when peak-shiftingphenomena are present, failing to produce physical-ly meaningful solutions. CombiFD (Ermon et al.2015) is another factor decomposition approach thatuses combinatorial constraints to simultaneouslyenforce some of the physical rules and accommodatepeak shifting, but requires solving a combinatorialproblem in each descent step, and is therefore com-putationally expensive and does not enforce all thephysics constraints.

Here, we describe Phase-Mapper, an AI platformfor rapidly solving the phase-mapping problem, inte-grating three key components: (1) cutting-edge AIsolvers, (2) human intelligence and feedback, and (3)high-throughput physical experiments. These com-ponents form an integrated process (see figure 1).

Phase-Mapper features novel solver AgileFD as akey component of the platform. Motivated by con-volutive NMF, AgileFD includes a set of lightweightupdating rules, and therefore a very fast gradientdescent process. AgileFD is flexible, allowing for theincorporation of additional contraints, as well ashuman feedback through refinement. AgileFD canalso run autonomously, producing physically mean-ingful solutions.

Phase-Mapper also provides tools for data explo-ration, visualization, and configuration that allowhuman experts as well as laypeople to analyze andimprove solutions.

Phase-Mapper’s solutions, obtained by the inter-action between solvers and human users orautonomously, can also shed light on the develop-ment of new physical experiments. For example, theresults can be incorporated into an active learningsystem, specifying regions of composition space tosample at higher resolution.

AgileFD: A Novel Phase-Mapping Solver

The Phase-Mapper platform features the AgileFDsolver for the phase-mapping problem. AgileFD usesiterative updates of candidate solutions that are sig-nificantly faster than previously proposed methods.Human experts can interact with the algorithm inreal time, and this speed is due to an efficient prob-lem representation. Let the XRD patterns for all sam-ples be represented by a matrix A, where each col-umn corresponds to one sample point and each rowcorresponds to Aj(q) for a particular value of q. Under

the assumptions of no noise and no shifting, mean-ing that λij = 1 for all i and j, describing A as a linearcombination of a few basis patterns Wi(q) is equiva-lent to factoring A as a product of two matrices Wand H:

(2)

Here, R denotes the approximate reconstruction of A.In this formulation, the columns of W form a set ofbasis patterns Wi(q), and the columns of H corre-spond to the values hij in equation 1. We enforcenonnegativity for W and H, which is required for thesolutions to be physically meaningful. Previousapproaches to solve the phase-mapping problembased on NMF have been unsuccessful in handlingpeak shifting, where λij ≠ 1. The first contribution ofAgileFD is to circumvent the shifting problem by alog space resampling. Under the variable transforma-tion q into log q, our signal becomes Wi(log q). Moreimportantly, the shifted phase Wi(log λq) becomesWi(log λ + log q), which transforms the multiplicativeshift in the q domain into a constant additive offset.This allows the problem to be formulated in terms ofconvolutive nonnegative matrix factorization. Afterthis variable substitution, we discretize the values ofallowed λ and interpolate the signals at the corre-sponding geometric series of q values. The problemcan then be written:

(3)

With the columns of W representing the basis pat-

A � W�m

m=1

M

� �Hm = R

A �W �H = R

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Figure 3. An Illustration of the Phase-Mapping Problem.

Given a material system with XRD data read at discrete points, find a set ofbasis phases, such that every point’s XRD data can be made by a linear com-bination of the basis phases. Here, the left image is the original data, theright image is the found basis phases, and the middle image represents howmuch of a particular phase (the “phase concentration”) is present at eachdata point along with the composition-dependent shifting.

1

2

XRD pattern at N composition samples

=

shiftphases +“copies”

K

conc.+

1 M

terns, W↓m is the result of shifting the rows of the Wmatrix down m rows, and filling the displaced rowswith zeros. This represents the basis patterns with aconstant offset in the log q domain, and is equivalentto the original multiplicative shift in the q domain.The columns of Hm act as the activation of basis pat-terns for the basis patterns shifted down m units.Note that when M = 1, this formulation is equivalentto NMF, aside from the log transformation.

We can think of AgileFD as a family of algorithmsthat can be adapted to use different loss functions,regularization, and certain imposed constraints.Equation 2 is adapted from convolutive NMF, whichwas first proposed to analyze audio signals(Smaragdis 2004). The phase-mapping problem dif-fers from previous applications of cNMF for blindsource separation as the log q domain is substitutedfor the time domain, and each source (phase) isexpected to appear at most once per sample with arelatively small offset. As in cNMF, AgileFD uses a gra-dient descent approach to fit W and H. When a gen-eralized Kullback-Leibler (KL) divergence is used inthe objective function, gradient updates can be writ-ten multiplicatively, and are applied iteratively untilconvergence. See Xue et al. (2017) for further details.

Lightweight Update Rules AgileFD’s linear gradient update rules solutions typi-cally converge within minutes. This is orders of mag-nitude faster than CombiFD, which uses a similarproblem formulation but with combinatorial con-straints explicitly enforced globally, using a mixed-integer programming (MIP) representation. Thisincreased efficiency of AgileFD enables high-through-put analysis and also makes it possible for a human tointeract with the system in almost real time.

Further Extensions of AgileFD for Materials Discovery The ultimate aim of the phase-mapping problem isto find a physically meaningful decomposition of thesignal. In the next few sections, we provide a numberof novel modifications to the basic AgileFD algo-rithm, in order to impose prior knowledge or addi-tional constraints derived from user interpretation ofa proposed solution.

AgileFD with Frozen ValuesIn the Phase-Mapper platform, the user is providedwith the opportunity to freeze individual values inthe W and H matrices. For example, a user mightspecify a known pattern or part of a previous solutionas a basis pattern a priori, freezing the correspondingrow or part thereof of W. Or the user might specifythat a certain set of samples contain only a singlephase and set the corresponding H values to zero. Theresult is interactive, iterative matrix factorization.

Custom Initialization By initializing basis patterns or coefficients to valuesclose to the expected solution, rather than randomvalues, the user can direct the search to the correctsolution space. We allow the user to specify basis pat-terns that can be taken from previous solutions, fromdata samples, or be provided manually, to use as aninitial value. Similarly, initial values for the activa-tion matrix can be specified.

Sparsity Regularization Sparse solutions are more consistent with the under-lying physics and are also usually more easily inter-preted. The Phase-Mapper system provides theoption to introduce a soft penalty term for sparsity inH, which can vary by index according to a humanexpert’s preferences.

The Gibbs Phase Rule In general, correct solutions to the phase-mappingproblem should follow the Gibbs phase rule, whichspecifies that the number of observed phases at a giv-en chemical composition is no more than the num-ber of chemical elements Nel:

(4)

Here, Ii,j is an indicator of whether phase i is presentat sample location j. Materials scientists might alsoknow a priori, or infer from previous proposed solu-tions, that certain regions contain fewer phases thanthe usual limit.

Such combinatorial constraints cannot be encod-ed directly in the update rules of AgileFD, which hasbeen used in previous methods such as CombiFD.However, these encodings result in a slow updateprocess, as we have to solve a MIP problem in eachiteration. As a novel routine, we apply the Gibbsphase rule by first solving the relaxed problem, thenchoose the best values to set to zero in H, and thenrefine the solution by applying the update rules untilconvergence. Because the update rules are multi-plicative, the zeroed values will remain zero.

The value to set to zero in H is independent foreach sample point j. This can be solved greedily if afaster solution is desired, or using a MIP formulation,which results in a small MIP program, or successiverounds of constraints and refinement, if a more pre-cise solution is desired with a not much longer waittime. In addition, in the presence of alloying, thenumber of possible phases is further reduced by one.This alloying rule can also be captured with a smallMIP program. These extensions are particularly use-ful when the unconstrained algorithm recovers asolution that is nearly correct except for relativelysmall violations of phase limits. See details in Bai etal. (2017).

Ii ,ji

� � Nel

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Phase-Mapper: A Human-Machine Integrated Platform

In this section, we present the Phase-Mapper plat-form workflow, which includes visualizing and ana-lyzing an instance file, setting the solver framework,analyzing the solution, and using that analysis toupdate the solver framework. The design objectiveswere simple: create a practical application that seam-lessly connects a visualization system with a power-ful solver that allows for interactive and large-scaleuse. The main features of Phase-Mapper are the visu-alization tools and the solver interface. The interfaceof the application can be seen in figure 4.

With Phase-Mapper, both the input materials sys-

tems and generated solutions can be visualized in thesame application. When an instance file of a materi-als system is uploaded to the system, the visualizerwill generate a composition map, which illustratesthe varying compositions of elements for all samplepoints. The user can freely inspect the XRD patternsof each sample point, as well as the heatmap of XRDpatterns for a slice of sample points. A slice heatmapexample is shown in the top left plot of figure 4,where a selected slice is indicated by the red datapoints. The heatmap plot on the top right of figure 4represents the XRD patterns of the sample points inthe slice.

When the solution files are loaded into the appli-cation, either uploaded by the user or generated by

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Figure 4. Screenshot of the Phase-Mapper Web Application.

Displayed in the top left is the Nb-Mn-VO system in composition space. Each dot corresponds to an XRD measurement. The dots high-lighted in red correspond to selected XRD measurements. The top right plot displays the heatmap of the selected data points’ XRD patterns,that is, it visualizes the XRDs for all red sample locations in the top left panel. The bottom left plot shows the basis phases found in theloaded solution of a particular data point. The bottom right plot shows the phase fields for the system.

the solver, two new plots are generated: (1) the basispatterns that were found as solutions and (2) a com-position map displaying the mixture proportions.

Connection to Solver The solving feature of Phase-Mapper enables users tointeract with the AI solver behind the scenes. Theuser can specify many solver parameters, such as howmuch to enforce sparsity, how many phases the solu-tion should have, and how much shift between basispatterns the solver should allow. The user can alsospecify initial or frozen values to use as basis patterns.Incorporating user inputs helps the solver improveefficiency and accuracy. We provide tools for expertusers to start the solver off closer to a solution, or todistort the solution space so the solver finds a moreaccurate solution.

Algorithm EffectivenessIn this section, we evaluate many solvers that wereproposed in recent years against our Phase-Mappersystem. We tested NMF (implemented as AgileFDwith M = 1), AgileFD, AgileFD with sparsity regular-ization (AgileFD-Sp), AgileFD with sparsity and theGibbs phase rule enforced (AgileFD-Sp-Gibbs), andCombiFD (Ermon et. al. 2015). We generated syn-thetic ternary metallic systems using data providedby the Materials Project (Jain et al. 2013), which pro-vides crystal structure information and energy of for-mation using density functional theory for eachphase. We applied a stylized model of solid solubili-ty and used structure interpolation to simulate mod-ified phase diagrams that include the additionaldegrees of freedom from alloying. We calculated XRDpatterns for each modified constituent, includingtheir interpolated structures, using pymatgen (Ong etal. 2013).

The quality of a solution is judged by how welleach sample’s reconstructed signal matches the cor-responding measured signal. We find the permuta-tion of the phases in the solution to best match theground truth. In general, we observe that the solu-tions found by AgileFD (including AgileFD-Sp andAgileFD-Sp-Gibbs) better match the ground truthwhen compared with NMF and CombiFD. NMFunderperforms because it cannot model peak shift-ing. Despite the fact that CombiFD also capturessome of the physical constraints, it does not scalewell because it formulates the physical constraintsusing mixed-integer programming. AgileFD withextensions (AgileFD-Sp and AgileFD-Sp-Gibbs) out-perform vanilla AgileFD. They are able to find solu-tions that better match the physical constraints.

Illustrative Example: Discovery of Nb-V-Mn Oxides Light Absorbers for Energy ApplicationsThe integration of the rapid solver with visualizationtools enables materials scientists to interact with the

data in a variety of ways. The web-accessible visuali-zation tools enable rapid data exploration by materi-als scientists, which empowers materials scientists toinject their expert knowledge into the solution, forexample by specifying the number of phases, theextent of alloying-based peak shifting, or the knownexistence of a phase in a certain composition region.In this way, Phase-Mapper can run in unsupervisedor semisupervised modes per the availability of priorknowledge. To demonstrate the phase-mapping capa-bilities and the importance of the Gibbs constraint,figure 5 contains solutions for the phase map of 317XRD patterns in the Nb-V-Mn oxide compositionspace using M = 10 shifted versions, which corre-sponds to approximately 2 percent alloying-basedpeak shifting. Although the phase behavior of bina-ry subcompositions (for example, Nb-V oxides) hasbeen previously studied, the ternary compositionsare being explored for the first time to discover solarlight absorbers for energy applications. Materialsresearchers were unable to obtain a meaningful phasediagram using manual analysis of this data set, evenwith advanced visualization tools, primarily becausethere are a number of phases with somewhat similarbasis patterns, and most basis patterns containdozens of peaks, yielding a collection of XRD pat-terns that are rich in information, but that exceedhuman conceptualization.

We show in the paper by Suram et al. (2016) thatwithout accounting for alloying-based peak shifting,solutions are not meaningful in a number of ways,most notably the basis patterns do not correspond toindividual phases because the intensity for a phasewhose patterns shift across the data set is spread outover multiple basis patterns, creating phase-mixedbasis patterns that are as difficult to interpret as themixed-phase patterns in the raw data. The overlap-ping features in the basis patterns amplify this prob-lem and result in its persistence even when alloying-based peak shifting is taken into account. When twophases have overlapping features in their basis pat-terns and the phases coexist in a range of composi-tions, approximately equal data reconstructions canbe obtained using basis patterns that each containone phase or that each contain a mixture of phases.To empower the algorithm to overcome this degen-eracy in phase map solutions, we additionally applythe Gibbs constraint on the number of phases thatcan coexist in each composition sample. Figure 5shows the basis patterns without (top) and with (bot-tom) application of this constraint, with one basispattern on top highlighted to show that it contains amixture of phases. So although this constraint isapplied on the activations of the basis patterns, itindirectly makes the basis patterns more physicallymeaningful. The Phase-Mapper solution also exhibitsexcellent composition space connectivity for eachphase concentration map, as expected for equilibri-um phase behavior, and it exhibits systematic com-

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positional variation in the shift parameter λ, demon-strating alloying within phases 3, 5, and 6. Incorpo-rating both an alloying-based peak shifting modeland the Gibbs phase constraint resulted in basis pat-terns and phase concentration maps that are physi-cally meaningful, which is emblematic of a generalstrategy for AI-enhanced scientific discovery —injecting scientific knowledge where possible hastrickle-down effects that result in scientifically mean-ingful solutions.

In addition to the broader implications of ouralgorithms for scientific discovery, the solutions infigures 5 and 6 are also emblematic of broader trendsin materials science. As new technologies are con-ceived, new materials are needed, and often thesematerials are required to simultaneously exhibit avariety of properties and perform a variety of func-tions. At JCAP, researchers are pursuing the discoveryof several materials, including photoanodes, whichare materials that must absorb sunlight and harnessits energy to oxidize water into oxygen, freeing pro-tons and electrons to be utilized in fuel synthesis.Metal oxides, such as the compositions in the Nb-V-Mn oxide composition library discussed earlier, areexcellent photoanode candidates because of theirgenerally good stability under these conditions.Among the challenges in identifying and tailoringmetal oxides to be effective photoanodes is the gen-eral difficulty in tuning their optical properties. Theband gap energy of a given material dictates therange of sunlight that can be utilized by the materi-al, and although materials with a variety of band

gaps are available in the photovoltaic and light-emit-ting diode fields, such band gap tuning is quite rarein metal oxides. Pattern 4 in figure 5 corresponds tothe MnV2O6 crystal structure, and as indicated by theactivation map for this phase, it exists over a range ofcompositions where the peaks shift due to alloying.By combining this data with band gap measure-ments, figure 6 shows that within this single phase,the band gap can be tuned from approximately 1.9 to2.1 eV. At the higher band gap range, the materials donot absorb orange or red light, but lowering the bandgap enables these absorptions and thus increases thepotential efficiency of the material. This discovery ofalloying-based band gap tuning in this crystal struc-ture is one component of the much broader portfo-lio of materials science needed to design and createphotoanode materials. More generally, the discoveryof a material for new technology is typically the cul-mination of a suite of smaller discoveries, and withAI algorithms such as those provided by Phase-Map-per, these discoveries are being accelerated and com-piled to create a more comprehensive understandingof the underlying science, thus changing the arc ofscientific discovery.

ConclusionIn this article, we show that the combination ofhigh-throughput experimentation, AI problem solv-ing, and human intelligence can yield rich scientificdiscoveries, with an application in materials science.A major, critically missing component of the high-

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Figure 5. Solutions for the Phase Map of 317 XRD Patterns in the Nb-V-Mn Oxide Composition Space.

The 317 XRD patterns measured in the Nb-V-Mn oxide composition space were analyzed to produce phase-mapping solutions. On top, thesix basis patterns obtained using AgileFD (blue) are shown along with the peak pattern (red sticks) for the phases identified by materials sci-entists. For pattern 2, this phase could be identified only after applying the Gibbs phase constraint, which, even though applied only to thebasis pattern activations, results in the procurement of more meaningful (phase-pure) basis patterns. The middle row shows the six basispatterns obtained with AgileFD using Gibbs phase constraints (black), with the bottom row showing the composition map of activationsof each basis pattern. There is a point for each of the 317 composition samples, with point size corresponding to the phase concentrationand color corresponding to the alloying-based peak shifting. Nonshifted (nonalloyed) samples show as black.

meanP1

K=6,

M=1

0(A

gile

FD)

K=6,

M=1

0(A

gile

FD_s

p_g

ibbs

)

P2 P3 P4 P5 P6

0.004

–0.00

0

B1 B2 B3 B3 B5 B6

throughput materials discovery pipeline is the abili-ty to rapidly solve the phase map identification prob-lem, which involves the determination of the under-lying phase diagram of a family of materials fromtheir composition and structural characterizationdata. To address this challenge, we developed Phase-Mapper, a comprehensive platform that tightly inte-grates XRD experimentation, AI problem solving,and human intelligence. The AI solvers in Phase-Mapper provide high-quality solutions to the phase-mapping problem within minutes. These solutionscan then be examined and further refined by materi-als scientists interactively and in real time. We havedeveloped a novel solver, AgileFD, that features light-weight iterative updates of candidate solutions and asuite of adaptations to the multiplicative updaterules. In particular, we have developed the ability toincorporate constraints that capture the physics ofmaterials as well as human feedback, enabling func-tionalities well beyond traditional demixing tech-niques and producing physically meaningful solu-tions. Phase-Mapper has been deployed at theDepartment of Energy’s Joint Center for ArtificialPhotosynthesis for materials scientists to solve a widevariety of real-world phase diagrams. Since thedeployment of Phase-Mapper, thousands of X-raydiffraction patterns have been processed and theresults are yielding the discovery of new materials forenergy applications, as exemplified by the discovery

of a new family of metal oxide solar light absorbers,among the previously unsolved Nb-Mn-V oxide sys-tem, which is provided here as an illustrative exam-ple. We believe Phase-Mapper will lead to furtherdevelopments in high-throughput materials discov-ery by providing rapid and critical insights into thephase behavior of new materials.

AcknowledgementsThis material is supported by NSF awards CCF-1522054 and CNS-0832782 (Expeditions), CNS-1059284 (Infrastructure), and IIS-1344201 (INSPIRE);and ARO award W911-NF-14-1-0498. Materialsexperiments are supported through the Office of Sci-ence of the U.S. Department of Energy under AwardNo. DE-SC0004993. Use of the Stanford SynchrotronRadiation Lightsource, SLAC National AcceleratorLaboratory, is supported by the U.S. Department ofEnergy, Office of Science, Office of Basic Energy Sci-ences, under Contract No. DE-AC02-76SF00515.

ReferencesBai, J.; Bjorck, J.; Xue, Y.; Suram, S.; Gregoire, J.; and Gomes,C. 2017. Relaxation Methods for Constrained Matrix Fac-torization Problems: Solving the Phase Mapping Problem inMaterials Discovery. In Proceedings of the 14th InternationalConference on Integration of Artificial Intelligence and Opera-tions Research Techniques in Constraint Programming, 104–

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Figure 6. Band-Gap Tuning.

The noted composition region (a) of the Nb-V-Mn oxide composition library contains high-phase concentration of phase 4 from figure 5.This basis pattern was matched to the MnV2O6 structure and in (b) the composition points are plotted using the alloying-based shift param-eter from the AgileFD with Gibbs constraint solution. This plot also notes the separately measured band gap energy of each sample, whichdetermines the amount of solar light that can be absorbed. As the material composition changes, the composition in this phase changesand causes the volume of the phase and the band gap to change. This type of “band gap tuning” is quite rare in metal oxides, and this dis-covery was enabled by the detailed phase map provided by Phase-Mapper.

b.a.

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1.005 1.007 1.009

2.0

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Junwen Bai is a PhD student in the Department of Com-puter Science at Cornell University. He received his bache-lor’s degree in computer science from Zhiyuan College,Shanghai Jiao Tong University.

Yexiang Xue is a PhD student in the Department of Com-puter Science at Cornell University. His research focuses ondeveloping intelligent systems that tightly integrate deci-sion making with machine learning and probabilistic rea-soning under uncertainty.

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Johan Bjorck is a PhD student in the Department of Com-puter Science at Cornell University. He received his bache-lor’s degree in engineering physics from Chalmers Univer-sity of Technology.

Ronan Le Bras is a research scientist at the Allen Institutefor Artificial Intelligence (AI2). His research interests includecomputational methods for large-scale combinatorial opti-mization, reasoning, machine learning, and human com-putation.

Brendan Rappazzo is a research assistant at the Institute forComputational Sustainability, in the Department of Com-puter Science at Cornell University. He received his BS inbioengineering and biomedical engineering from Universi-ty of Maryland, College Park.

Richard Bernstein is a research programmer/analyst at theInstitute for Computational Sustainability, in the Depart-ment of Computer Science at Cornell University, and theproject coordinator and IT manager for the ComputationalSustainability Network (CompSustNet).

Santosh K. Suram is a research scientist IV at ToyotaResearch Institute and a member of the Computational Sus-tainability Network. Suram’s expertise is in combiningmaterials informatics and high-throughput experimenta-tion for accelerated materials discovery, specifically for ener-gy materials. Suram strongly believes that AI will play a piv-otal role in accelerating materials discovery.

R. Bruce van Dover is the Walter S. Carpenter, Jr., professorof engineering and director of the Department of MaterialsScience and Engineering at Cornell University, a seniormember of the IEEE, and a fellow of the American PhysicalSociety. His research currently focuses on the synthesis andcharacterization of thin films, and on developing and usinghigh-throughput techniques for the discovery of ionic con-ductors, fuel cell catalysts, and other energy-related materi-als.

John Gregoire leads the High Throughput Experimentationgroup at the California Institute of Technology. He is alsothe thrust coordinator for photoelectrocatalysis in the JointCenter for Artificial Photosynthesis and an associate direc-tor in the Computational Sustainability Network. Gregoireis an expert in the discovery of energy-related materials anda strong advocate for further adoption of AI in materialsresearch.

Carla P. Gomes is a professor of computer science at CornellUniversity and the director of the Cornell Institute for Com-putational Sustainability. Gomes is the lead PI of an NSFExpeditions in Computing, establishing and nurturing thenew field of computational sustainability. She is a fellow ofAAAI, AAAS, and ACM.

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