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Phase-Mapper: An AI Platform toAccelerate High Throughput Materials Discovery
Yexiang XueDepartment of Computer Science
Cornell [email protected]
Junwen BaiZhiyuan College
Shanghai Jiao Tong University, Chinabjw [email protected]
Ronan Le Bras, Brendan RappazzoDepartment of Computer Science
Cornell University{rl454, bhr54}@cornell.edu
Richard Bernstein, Johan Bjorck, Liane LongpreDepartment of Computer Science
Cornell University{rab38, njb225, lfl42}@cornell.edu
Santosh K. SuramJoint Center for Artificial Photosynthesis
California Institute of [email protected]
Robert B. van DoverMaterials Science and Engineering
Cornell [email protected]
John GregoireJoint Center for Artificial Photosynthesis
California Institute of [email protected]
Carla P. GomesDepartment of Computer Science
Cornell [email protected]
AbstractHigh-Throughput materials discovery involves the rapid syn-thesis, measurement, and characterization of many differentbut structurally-related materials. A key problem in materi-als discovery, the phase map identification problem, involvesthe determination of the crystal phase diagram from the ma-terials composition and structural characterization data. Wepresent Phase-Mapper, a novel AI platform to solve the phasemap identification problem that allows humans to interactwith both the data and products of AI algorithms, includ-ing the incorporation of human feedback to constrain or ini-tialize solutions. Phase-Mapper affords incorporation of anyspectral demixing algorithm, including our novel solver, Ag-ileFD, which is based on a convolutive non-negative matrixfactorization algorithm. AgileFD can incorporate constraintsto capture the physics of the materials as well as human feed-back. We compare three solver variants with previously pro-posed methods in a large-scale experiment involving 20 syn-thetic systems, demonstrating the efficacy of imposing phys-ical constrains using AgileFD. Phase-Mapper has also beenused by materials scientists to solve a wide variety of phasediagrams, including the previously unsolved Nb-Mn-V oxidesystem, which is provided here as an illustrative example.
1 IntroductionThe wonders of modern technology can largely be attributedto advances in materials science that enable innovationsfrom semiconductors to renewable energy. High throughputmaterials discovery comprises a suite of emerging method-ologies to rapidly identify new materials, especially, un-discovered materials that are critical for next-generationtechnologies (Green, Takeuchi, and Hattrick-Simpers 2013).Specifically, this method generates 102-103 unique materialsand forms them into a “library” and then rapidly screens forthe properties of interest.
To analyze the vast amount of data that are generated inhigh throughput experiments, automatic analysis becomes
Copyright c© 2017, Association for the Advancement of ArtificialIntelligence (www.aaai.org). All rights reserved.
Figure 1: The Phase-Mapper Platform integrates experimen-tation,AI solvers and human feedback into a platform forHigh Throughput Materials Discovery for discovering newmaterials.
imperative. The traditional workflow in materials sciencerelies on iterative manual analysis and heuristics, resultingin months or years of analysis for a single physical system.This quickly becomes a bottleneck as humans are unable tokeep up with the rate at which data is generated from highthroughput experiments. The need for automatic and scal-able tools in materials science provides computer scientistsunique opportunities to apply cutting edge techniques in Ar-tificial Intelligence and data science to accelerate the mate-rials discovery process.
In this paper we address the phase mapping problem,a central issue in high-throughput materials discovery, andprovide an efficient method of solving it which is cur-rently a critically missing component in the high through-put materials toolbox. A materials phase is a fundamen-tal property of a material composition (a mixture of el-ements) that describes the arrangement of the constituent
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atoms. X-ray diffraction (XRD) is a ubiquitous techniqueto characterize phases, as it produces a signal containing aseries of peaks that serve as a “fingerprint” of the under-lying atomic arrangement (or crystal structure). Using tra-ditional methods, materials scientists can obtain and inter-pret 1-10 XRD measurements per day, and with the recentdevelopment of automated, synchrotron-based XRD exper-iments, the measurement throughput has been acceleratedto 103-105 measurements per day (Gregoire et al. 2009;Gregoire et al. 2014). The creation of a phase mapping algo-rithm that generates phase diagrams from these data remainsan unsolved problem in materials science despite a series ofadvancements over the past decade (Hattrick-simpers, Gre-goire, and Kusne 2016). The most pertinent need is to gen-erate a physically-meaningful phase diagram for the 102-103 materials in a given library, which relies on the spectraldemixing of the 102-103 XRD patterns into a small set ofbasis patterns (typically less than 10).
To address this substantial challenge, we developedPhase-Mapper, a comprehensive AI platform that tightlyintegrates XRD experimentation, AI problem solving, andhuman intelligence for the phase mapping problem (See Fig-ure 1). In this platform, within minutes, AI solvers giveapproximate results for the phase mapping problem, whichthen are examined and further refined by materials scien-tists interactively and in real time. In addition, the results ofPhase-Mapper can be used to further inform future experi-mental designs. The demixing algorithm is a cornerstone ofthe Phase-Mapper platform, and we have developed a novelsolver called AgileFD, based on Convolutive Non-negativeMatrix Factorization (cNMF), a method that has been ap-plied to blind source separation of audio signals and speechrecognition (Smaragdis 2004; Mørup and Schmidt 2006).
AgileFD features a lightweight iterative updates ofcandidate solutions, and we have developed a suite of adap-tations that enable functionalities beyond cNMF. The exten-sions for AgileFD described here include incorporation ofconstraints to encode both human input, that capitalizes ona researchers knowledge of a particular dataset, and a prioriknowledge of the problem related to the underlying physicsof phase diagrams.This, as demonstrated below, can be crit-ical in obtaining physically meaningful solutions. In devel-oping the Phase-Mapper platform, careful attention has beengiven to deliver a rich suite of capabilities while maintain-ing solver convergence times within minutes, which enablesresearchers to interact with the solver to refine the solution.
We compare three variants of the AgileFD with previ-ously proposed solvers on an experiment involving 20 syn-thetically generated systems. Our results show that Ag-ileFD outperforms previous solver in terms of reconstruc-tion error and model correctness, owing to the fact that themodel represents peak shifting in an efficient way. In addi-tion, light-weight update rules allow AgileFD to convergemore quickly than previous solvers and with its extensionsthe obtained solutions tend to be more physically meaning-ful.
Phase-Mapper has been deployed in the Joint Center forArtificial Photosynthesis (JCAP) supported by the Depart-ment of Energy for materials scientists to solve a wide va-
High-ThroughputSynthesis
Metal1Metal3Metal2
SiliconWafer
High-ThroughputCharacterization
NationalX-raySources
Beam
Intensity
Diffractionangle
X-RayDiffraction
Figure 2: High throughput synthesis and characterization ofmaterials.
riety of phase diagrams, including the previously unsolvedNb-Mn-V oxide system, which is provided here as a casestudy and an illustrative example of the importance of en-coding physical constraints to obtain physically-meaningfulphase diagram solutions.
2 Phase-Mapper: AI for Materials DiscoveryHigh throughput materials discovery is an experimentationpipeline for rapidly synthesizing and identifying new mate-rials. In this pipeline, a handful of elements are depositedtogether on a two-dimensional substrate, so that different lo-cations on the substrate receive varying proportions of theelements. This variation in elemental composition across thesubstrate gives rise to the forming of a discrete set of mate-rials. Each material is present on particular segments of thesubstrate, due to the continuous change of the proportion ofthe elements across the substrate.
After deposition, sample locations on the substrate areprobed with high energy X-rays, and because each materialhas a characteristic XRD pattern it becomes possible to char-acterize the discrete set of materials present on the substrate.Figure 2 provides a graphical illustration of the synthesis andcharacterization step in the high throughput pipeline.
Unlike in traditional powder diffraction methods for ma-terials discovery, the XRD patterns obtained in the highthroughput pipeline can be composed of the characteristicXRD patterns of several materials. Therefore, the phase-mapping problem, which is to identify the characteristicXRD patterns for the materials (or basis patterns) that de-mix the signal, lies at the heart of the analysis of highthroughput data. Mathematically, the measured XRD patternin the j-th sample point can be characterized by a one di-mensional signal Aj(q). The “scattering vector magnitude”q is a monotonic transformation of the diffraction angle,and is directly related to the spacing of atoms in a crystal.The phase-mapping problem is to find a small number ofphases W1(q), . . . ,WK(q), such that the XRD patterns ateach sample point can be explained by a linear combinationof phases:
Aj(q) ≈K∑i=1
hijWi(λijq). ∀j (1)
In the above definition, we write Wi(λijq) to allow for thephases to scale slightly according to parameter λij at eachsample point. This is the result of a commonly observedform of alloying, a process that can typically be approxi-mated by a multiplicative scaling of the XRD pattern of aspecific phase in the q domain. We also call this process peak
shifting, because the effect appears to make XRD patterns inthe data shift to the left or to the right. In addition to the com-plications introduced by peak shifting, there are a number ofother constraints on the solution of the phase-mapping prob-lem, arising from the fact that the solution must describe asystem constrained by the laws of physics. The most promi-nent is the so called Gibbs phase rule, which states that nomore than three coefficients among hij for fixed j may benonzero (in a ternary system). Additionally, how the hij mayvary spatially on the substrate, as well as the shapes that Wi
may take, are constrained by physical laws.Fundamentally novel techniques are required to solve the
phase mapping problem quickly and accurately. A num-ber of automatic techniques have been developed in re-cent years, which can be broadly grouped into clustering,constraint reasoning, and factor decomposition approaches.Proposed clustering methods such as hierarchical cluster-ing (HCA) (Long et al. 2007), dynamic time warping kernelclustering (Le Bras et al. 2011) and mean shift theory (Kusneet al. 2014) produce maps of phase regions, but fail to re-solve mixtures or identify basis patterns, and do not nec-essarily produce results consistent with physics. Constraintreasoning approaches include satisfiability modulo theory(SMT) methods (Ermon et al. 2012), which can providephysically meaningful results, but depend heavily on effec-tive pre-processing such as peak identification, and are com-putationally intensive. Approaches based on non-negativematrix factorization (NMF) (Long et al. 2009) are computa-tionally efficient, but generally perform poorly when peak-shifting phenomena are present. Another factor decompo-sition approach called CombiFD (Ermon et al. 2015) usescombinatorial constraints to simultaneously enforce physi-cal rules and accommodate peak shifting, but requires solv-ing a combinatorial problem in each descent step, and istherefore computationally expensive.
We propose the AI platform Phase-Mapper, which inte-grates three key components to solve the problem: (i) cuttingedge AI solvers; (ii) human intelligence and feedback; and(iii) high-throughput physical experiments (See Figure 1).
• Our platform is supported by cutting edge AI solvers, in-cluding Non-negative Matrix Factorization (Lee and Se-ung 2001), and CombiFD. We also highlight a new solvercalled AgileFD as a key component of the platform. Mo-tivated by Convolutive NMF, AgileFD features a set oflight-weight updating rules and therefore a very fast gra-dient descent process. AgileFD is also flexible, allowingfor additional physical constraints or incorporation of hu-man feedback through rounding and refinement.
• Our platform also provides tools for data exploration, vi-sualization, and configuration that allows human expertsas well as laypeople to analyze and improve solutions.
• The solutions obtained by the interaction between solversand human users could also shed light on the developmentof new physical experiments, for example by specifyingregions of composition space to sample at higher resolu-tion (active learning). Therefore, the three components ofPhase-Mapper form an integrated process (see Figure 1).
3 AgileFD as a Novel SolverAgileFD is a key component in the Phase-Mapper platform.Compared with previously proposed methods for solvingthe phase-mapping problem, AgileFD features quick itera-tive updates of candidate solutions, which makes it possi-ble for human experts to interact with the algorithm in realtime. The key behind this speed lies in the efficient problemrepresentation. Let the XRD patterns for all samples be rep-resented by a matrix A where each column corresponds toone sample point and each row corresponds to Aj(q) for aparticular value of q. Under the assumptions of no noise andno shifting, i.e. for all i, j, λij = 1, describing A as a lin-ear combination of a few basis patterns Wi(q) is equivalentto factorizing A as a product of two low rank matrices Wand H . We enforce nonnegativity for W and H , which isrequired for the solutions to be physically meaningful.
A ≈W ·H.In this formulation, the columns of W form a set of basis
patterns Wi(q), while the columns of H corresponds to thevalues hij in equation 1. Previous approaches to solve thephase-mapping problem based on NMF have been unsuc-cessful in handling peak shifting, i.e. λij 6= 1. The first con-tribution of AgileFD is to circumvent the shifting problemby a log space resampling. Under the variable substitutionq → log q our signal becomes Wi(log q). More importantly,the shifted phase Wi(log λq) becomes Wi(log λ + log q),which transforms the multiplicative shift in the q domaininto a constant additive offset. Once a solution is found, thetransformation is reversed, and the solution is interpolated atthe original q values. This allows the problem to be formu-lated in terms of convolutive nonnegative matrix factoriza-tion. After this variable substitution, we discretize the valuesof allowed λ and interpolate the signals at the correspondinggeometric series q values. The problem can then be written:
A ≈ R =∑m
W ↓m
·Hm. (2)
Here, the columns of W still represent basis patterns. W ↓mis the result of shifting the rows of the W matrix down mrows, and padding the shifted m rows with 0, representingthe basis patterns with a constant offset in the logq domain,which is equivalent to the original multiplicative shift in theq domain. The columns of Hm act as the activation of basispatterns for the basis patterns shifted down m units. Notethat when M = 1, this formulation is equivalent to NMFaside from the log transformation.
AgileFD is a family of algorithms, which can be adaptedto use different loss functions, regularization, and certain im-posed constraints. Equation (2) is adapted from ConvolutiveNMF (cNMF), which was first proposed to analyze audiosignals (Smaragdis 2004). The phase-mapping problem dif-fers from previous applications of cNMF for blind sourceseparation as the logq domain is substituted for the time do-main, and each source (phase) is expected to appear at mostonce per sample with a relatively small offset. As in cNMF,we use a gradient descent approach to fitW andH . The gra-dient updates can typically be written multiplicatively, and
are applied iteratively until convergence. For example, whenthe generalized Kullback-Leibler divergence is chosen as theloss function, the following update rules can be applied:
Hm = Hm ◦ (W↓m)T (A/R)
(W ↓m)T1
, (3)
W =W ◦
∑m
(A↑m
R↑m
)(Hm)
T
∑m
1 (Hm)T
. (4)
Here, A↑m denotes A with all rows shifted upwards m stepsand padded with zeros, and 1 is the ones matrix. All divi-sions the in above expressions are taken elementwise, and ◦denotes the Hadamard product.
Lightweight Update Rules AgileFD’s gradient descentmethod with simple linear update rules results in very fastconvergence, typically reaching local minima within min-utes or faster. This runtime performance is orders of magni-tude faster than CombiFD, which uses a similar problem for-mulation but with combinatorial constraints enforced. Thisincreased efficiency enables high throughput analysis andalso makes it possible for a human to interact with the sys-tem almost in real time.
Further Extensions of AgileFD for Materials DiscoverySince the ultimate aim of the phase-mapping problem is tofind a physically meaningful decomposition of the signal, forwhich the loss function is just a proxy, we must allow for ex-perts to modify and inspect any candidate solution. It is notfeasible to encode all physical constraints or the knowledgeof a materials scientist within the solver a priori. Therefore,in the next few sections, we provide a number of novel mod-ifications to the basic AgileFD algorithm, in order to imposeprior knowledge or additional constraints derived from userinterpretation of a proposed solution.
AgileFD with Frozen Values In the Phase-Mapper plat-form, the user is provided with the opportunity to freeze in-dividual values in the W and H matrices. For example, auser might specify a known pattern or part of a previous so-lution as a basis pattern a priori, freezing the correspondingrow or part thereof inW to a known value. Or the user mightspecify that a certain set of samples contain only a single ma-terial phase and set the non-corresponding H vales to zero.The result is an interactive, iterative matrix factorization. Weadapt the update rules to the this new scenario.Theorem 3.1. When the generalized KL Divergence is cho-sen as the loss function, the objective is always non-negativeand non-increasing when applying the update rules in Equa-tion 3 and Equation 4 while holding one or more basis pat-terns in W or phase proportions in H fixed.
The proof of Theorem 3.1 can be found in (Xue et al.2016). It relies on the fact that the auxilliary function usedto prove that the normal update rules are non-increasing hascertain linearity properties.
Custom Initialization The local optimum to which Ag-ileFD converges is not guaranteed to be the correct opti-mum. In this case, initializing basis patterns or coefficients
to values close to the expected solution, rather than randomvalues, the user can direct the search to the correct solutionspace. We allow the user to specify basis patterns that canbe taken from previous solutions, data samples, or providedmanually, to use as an initial value. Similarly, initial valuesfor the activation matrix can be specified.
Sparsity Regularization Sparse solutions are usuallymore easily interpreted, and in materials science they aremore likely to be consistent with the underlying physics.The AgileFD system provides the option to introduce a spar-sity term in H which can vary by index according to anhuman experts preferences. Using L1-regularization for Hand sparsity weight matrices γm, the sparse generalized KL-divergence objective function becomes:∑
i,j
(Ai,j log
Ai,j
Ri,j−Ri,j +Ai,j
)+∑m
‖γm ◦Hm‖1.
The corresponding update rule for H becomes:
Hm = Hm ◦ (W ↓m)T (A/R)
(W ↓m)T1+ γm
.
In order to avoid the degenerate solution where H → 0,each basis pattern of W is L2-normalized at the beginningof each update iteration.
Sparsity Constraints In general, correct phase map so-lutions should follow the Gibbs phase rule, which specifiesthat under certain experimental conditions, the number ofobserved phases at a given chemical composition is less thanor equal to the number of chemical elements Nel present:∑
i
Iij ≤ Nel. (5)
Here, Iij is an indicator of whether phase i is present at sam-ple location j. Materials scientists might also know a priori,or infer from previous proposed solutions, that certain re-gions contain fewer phases than the usual limit.
Such combinatorial constraints cannot be encoded di-rectly in the update rules of AgileFD. One way to enforcethese constraints, which has been used in previous methodssuch as CombiFD (Ermon et al. 2015), is to directly encodeit as hard combinatorial constraints. However, this results ina very slow update process, as we have to solve a Mixed In-teger Programming (MIP) problem in each iteration. As anovel routine, we apply the Gibbs Phase rule by first solvingthe relaxed problem, then we choose the best values to set tozero in H to satisfy Equation 5, and then refine the solutionby applying the update rules until convergence. Because theupdate rules are multiplicative, the zeroed values will remainzero.
Choosing the values to zero in H is independent for eachsample point j. This can be solved greedily if a faster solu-tion is desired, or using a MIP formulation, and/or succes-sive rounds of constraints and refinement, if a more precisesolution is desired with a somewhat longer wait time. Thisextension is particularly useful when the unconstrained al-gorithm recovers a solution that is nearly correct except forrelatively small violations of phase limits.
Figure 3: Two heatmaps of XRD patterns generated by tak-ing slices in the visualizer.
4 Human Integrated PlatformWe provide an integrated workflow with the Phase-Mapperplatform, which includes visualizing and analyzing an in-stance file, setting the solver framework, analyzing the solu-tion and using that analysis to update the solver framework.The design objectives were simple: create a practical appli-cation that seamlessly connects a visualization system witha powerful solver, that could be utilized in a large scale man-ner. The main features of Phase-Mapper are the visualizationtools and the solver interface.
Visualizer Phase-Mapper provides a way to visualize boththe input data as well as the solution that is generated.When an instance file of a materials system is uploaded tothe system, the visualizer will generate a composition map,which illustrates the varying compositions, of elements, forall sample points. The user can freely inspect the XRD pat-terns of each sample point, as well as the heatmap of XRDpatterns for a slice of sample points. A slice heatmap exam-ple is shown in Figure 3, where two selected slices of sam-ple points are shown in grey. The heatmap on the left repre-sents the XRD patterns at the sample points in the slice. Theheatmap also allows for multiple views of the data, empha-sizing different parts of the XRD signal, as well as a way tofreeze the current heatmap.
When the solution files are loaded to the application, ei-ther uploaded by the user or generated by the built in solver,several new plots are generated. The first of which, plots thebasis patterns that were found as solutions. The second isanother composition map, displaying how much each ba-sis pattern contributed to each sample point’s signal. Thethird plot shows the actual XRD signal as well as the re-constructed signal for a user specified sample point. Addi-tionally the user can toggle that plot so it shows each basispattern’s contribution to the reconstructed signal.
Connection to Solver The solving feature of Phase Map-per enables users to interact with the AI solver behind thescenes. The user can specify many solver parameters suchas how much to enforce sparsity, how many phases the so-lution should have, and how much shift between basis pat-terns it should allow. Perhaps the most useful parameter thatcan be set is the initilization or freezing of a particular sam-ple point’s XRD pattern as a phase in the actual solution.The user can select a single data point on the compositionmap and use its XRD pattern as an initialization basis forthe solver, or freeze that pattern as one of the basis patterns.The former is ideal for when the user believes a data point’sXRD pattern is similar to a basis pattern solution. The latter
is best for when the user believes a data point’s XRD pat-tern is a basis pattern solution. Both input parameters aimto help the solver more efficiently and accurately find a so-lution, either by starting the solver off closer to a solution,or distorting the solution space so the solver finds a moreaccurate solution.
Infrastructure It was pertinent that this tool be highly ac-cessible, and for this reason a web application was chosen asthe medium to deploy this project. This application runs onany modern web browser and for that reason is completelycross-platform. On the front end, HTML and CSS were usedto generate the aesthetics and structure for how the differentcomponents were presented to the user. JavaScript handledall of the client side visualization and functionality. Jqueryand Ajax were used to handle the client-server communica-tion, and all server side functionality was given by PHP. Thesolution parameters were passed to the solver using PHP andsolver itself was written in c++. Little maintence of the plat-form is needed as the methods in materials science as notexpected to change, and as the intended audience is wellversed in the phase mapping problem there is scarce needfor tutorials.
5 Experiments5.1 Large Scale ExperimentsDespite several solvers having been proposed (Le Bras et al.2011; Long et al. 2007; Kusne et al. 2014; Ermon et al. 2012;Ermon et al. 2015) in recent years, most solvers were con-figured and tested only on a handful of systems. As one con-tribution of this paper, we provide a large-scale evaluationof various solvers on the phase-mapping problem.
We generated synthetic ternary metallic systems usingdata provided by the Materials Project (Jain et al. 2013),which provides theoretical crystal structure information us-ing Density-Functional Theory based convex hull construc-tion in composition, energy space that predicts compositionsand corresponding lowest energy of formation atomic con-figurations that form the vertices of the convex hull.
Ac-A
g-Hg
Ag-B
a-Sn
Au-B
i-Os
Au-C
d-M
gBi
-Cu-
TcCd
-Hg-
LuCu
-Ru-
TmEr
-Ga-
OsFe
-Mg-
NbGa
-V-Z
rHf
-Ir-P
dIn
-Nb-
PtIr-
Tm-Y
La-R
u-Sn
Li-Pd
-Tc
Lu-R
h-Tl
Lu-R
u-Sc
Mg-
Os-S
cM
g-Ru
-Ta
Pd-S
c-Tc
0.00
0.05
0.10
0.15
0.20
0.25
Norm
alize
d L2
Los
s
NMFAgileFDAgileFD_SpAgileFD_Sp_GibbsCombiFD
Figure 4: Normalized L2 Loss vs. ground truth phases forNMF, AgileFD, AgileFD with sparsity, AgileFD with spar-sity & Gibbs phase rule and CombiFD for 20 physical sys-tems. AgileFD, AgileFD with sparsity, AgileFD with spar-sity & Gibbs phase rule perform best.
Ac-A
g-Hg
Ag-B
a-Sn
Au-B
i-Os
Au-C
d-M
gBi
-Cu-
TcCd
-Hg-
LuCu
-Ru-
TmEr
-Ga-
OsFe
-Mg-
NbGa
-V-Z
rHf
-Ir-P
dIn
-Nb-
PtIr-
Tm-Y
La-R
u-Sn
Li-Pd
-Tc
Lu-R
h-Tl
Lu-R
u-Sc
Mg-
Os-S
cM
g-Ru
-Ta
Pd-S
c-Tc
0500
1000150020002500300035004000
Tim
e co
st(s
)
NMFAgileFDAgileFD_SpAgileFD_Sp_GibbsCombiFD
Figure 5: Runtime for NMF, AgileFD, AgileFD with spar-sity, AgileFD with sparsity & Gibbs phase rule and Comb-iFD to solve 20 physical systems. Times for CombiFD arefor up to 15 iterations. The times for the other solvers areuntil convergence (convergence gap 2× 10−5). A time limitof 1 hour was imposed for all solvers.
The XRD patterns at compositions between the verticesof the convex hull are calculated, using pymatgen (Ong etal. 2013), by interpolation of a) XRD patterns of the ver-tex phases or b) modified XRD patterns of the vertex phasesto accommodate alloying. We applied a stylized model ofsolid solubility and alloying, and used structure interpola-tion to simulate modified phase diagrams that include theadditional degrees of freedom from alloying. We calculatedXRD patterns for each modified constituent, including theirinterpolated structures, and combined them according to themixture proportions in the phase diagram.
We selected 20 examples containing varying numbers ofphases and amounts of alloying for our experminent. Thesedata reflect many properties of real experimental data in-cluding physically plausible combinations of basis XRD pat-terns, connectivity relationships of phase fields, adherence toGibbs phase rule, and peak shifting. These data also provideground truth information, so that we can directly compareour solution models to those that generated the data.
We compare the resulting phases found by differentsolvers against the ground truth phases, which is knownbeforehand from the simulation used to generate these 20systems. We tested NMF1, AgileFD, AgileFD with SparsityRegularization, AgileFD with Sparsity and Gibbs phase ruleenforced, and CombiFD. The convergence gap for NMF andAgileFD are set to be 2 × 10−5, the MIP gap of CombiFDis set to be 0.2, and the sparsity parameter is set to 0.35 inAgileFD. All solvers were given a time limit of 1 hour, andallocated one compute core.
First we evaluate the solution quality by comparing themodeled signal for each phase at each sample point, includ-ing shift, to the known signal from that phase at that samplepoint. We find the permutation of the phases in the solu-
1For consistency, we implemented NMF using generalized KL-divergence loss, equivalent to AgileFD with M = 1. Because ofthis, and the resulting use of log scaling of q, this implementationof NMF differs from that previously applied to the phase mappingproblem (Long et al. 2009).
System K NMF AgileFD AgileFD AgileFD CombiFDSp Sp Gibbs
Ac-Ag-Hg 5 0.35 0.28 0.43 1.00 0.00Ag-Ba-Sn 13 0.09 0.09 0.21 1.00 0.00Au-Bi-Os 4 0.81 0.77 0.81 1.00 0.57Au-Cd-Mg 12 0.12 0.12 0.20 1.00 0.38Bi-Cu-Tc 3 1.00 1.00 1.00 1.00 1.00Cd-Hg-Lu 7 0.12 0.11 0.22 1.00 0.65Cu-Ru-Tm 6 0.32 0.27 0.62 1.00 0.59Er-Ga-Os 8 0.08 0.12 0.43 1.00 0.31Fe-Mg-Nb 4 0.73 0.78 0.89 1.00 0.85Ga-V-Zr 13 0.10 0.07 0.41 1.00 0.51Hf-Ir-Pd 8 0.27 0.38 0.51 1.00 0.43In-Nb-Pt 10 0.14 0.14 0.34 1.00 0.00Ir-Tm-Y 4 0.86 0.94 0.98 1.00 0.54
La-Ru-Sn 12 0.12 0.09 0.32 1.00 0.63Li-Pd-Tc 8 0.25 0.48 0.49 1.00 0.53Lu-Rh-Tl 9 0.17 0.12 0.36 1.00 0.00Lu-Ru-Sc 4 0.51 0.95 0.95 1.00 0.00Mg-Os-Sc 7 0.38 0.35 0.57 1.00 0.45Mg-Ru-Ta 8 0.22 0.20 0.27 1.00 0.00Pd-Sc-Tc 9 0.24 0.14 0.44 1.00 0.38
Table 1: Percentage of sample points that satisfy the GibbsPhase Rule for each solver on 20 physical systems. Phasesthat account for less than 1% of the modeled signal are notcounted towards this phase limit.
tion to best match the ground truth, and calculate the L2 lossfor each component. These are summed over all phases andsamples, and scaled by the total value of all signals in theexample system.
As shown in Figure 4, in general the solutions foundby AgileFD (including AgileFD-sparsity and AgileFD-sparsity-gibbs) better match the ground truth when com-pared with NMF and CombiFD. NMF underperforms be-cause it cannot model peak shifting. CombiFD often onlycompleted a few iterations within the time limit, and in afew cases timed out in the first iteration. These timeouts be-fore convergence resulted in lower solution quality. Figure 5compares the runtimes on each instance.
AgileFD with sparsity regularization and/or Gibbs phaserule enforced outperforms AgileFD without extensions.They are able to find solutions that better match the physi-cal constraints. We calculate the percentage of sample pointsthat satisfy the Gibbs phase rule, for the solutions found byeach solver for each instance. The result is shown in Ta-ble 1.When sparsity regularization (or, by design, the Gibbsphase rule) is enforced, the solutions tend to better satisfythe Gibbs phase rule.
5.2 Case Study: Nb-V-Mn OxidesThe integration of the rapid solver with visualization toolsenables materials scientists to take advantage of the tunableinitialization and constraint parameters to create a meaning-ful solution. An illustrative example is found in a ternarycomposition library containing a broad range of composi-tions in the Nb-V-Mn oxide space. While the phase behaviorof binary sub-compositions (e.g. Nb-V oxides) have beenpreviously studied, the ternary compositions are being ex-plored for the first time to discover solar light absorbers for
λ−λmean
B1 B2 B3 B4 B5 B6
P1 P2 P3 P4 P5 P6
K=6
, M=1
0(A
gile
FD)
K=6
, M=1
0(A
gile
FD_s
p_gi
bbs)
Figure 6: TheK=6 basis patterns for the AgileFD solution (top) and AgileFD-sparsity-gibbs solution (middle) for the Nb-V-Mnsystem are shown along with stick patterns (translucent red) of the crystal structures identified using the AgileFD-sparsity-gibbssolution. The primary discrepancy in the AgileFD solution is highlighted in yellow. The phase map representation ofH is shownas a series of composition plots for the AgileFD-sparsity-gibbs solution (bottom). For each basis phase, the composition plotshows the samples containing the phase with point size indicating the concentration of the phase and the point color indicatingthe fractional shift with respect to the plotted basis patterns. The elemental labels for the composition diagrams are shown inFigure 7.
K=6, M=1 (NMF) K=6, M=10(AgileFD)
K=6, M=10(AgileFD_sp_gibbs) Num.
phases
Figure 7: Composition maps of the number of theK=6 basiscomponents utilized in the solutions from 3 different solvers:NMF, AgileFD, and AgileFD-sparsity-gibbs. Values in ex-cess of 3 are non-physical.
energy applications. While the set of 317 XRD patterns pro-vides sufficient information to solve the phase behavior ofthese oxides, the materials researchers were unable to obtaina meaningful phase diagram for over a year due to the com-plex phase behavior that evaded comprehension via man-ual analysis. Using the visualization tools and rapidly tryingvarious solutions, the researchers determined that there areK=6 primary phases in this dataset. For this case study, wecompare solutions from 3 different solvers with K=6: (1)NMF; (2) AgileFD with M=10, which includes shifting ofeach basis pattern by up to approximately 2%; (3) AgileFD-sparsity-gibbs with M=10.
As described above, adherence to the Gibbs phase ruleis important for providing the researcher with a meaning-ful phase diagram. The number of basis patterns utilized foreach sample is shown in Figure 7 for each of the 3 solu-tions. The NMF solution adheres to the Gibbs phase rule foronly 4% of the samples, with nearly half of the samplesutilizing all 6 basis patterns. While the AgileFD solution ismore meaningful than the NMF solution due to the trackingof alloying via basis pattern shifting, the AgileFD solutionuses more phases for several compositions, corresponding to
a worse violation of Gibbs phase rule. This property of theAgileFD solver may be understood intuitively by consider-ing that for a given sample, any of the M copies of eachbasis pattern can be utilized to model small features in theXRD patterns, resulting in small amounts of shifted patternsto appear across the composition space and further motivat-ing the encoding of Gibbs phase rule in the solver. With thisconstraint, the AgileFD solution utilizes a maximum of 3phases and in some composition regions only 2 phases areutilized.
Figure summarizes the AgileFD-sparsity-gibbs solution,and the desirable attributes that were not enforced but pro-vide meaning to the researcher include: (1) excellent match-ing of the primary peaks of each basis pattern with a knowncrystal structure, demonstrating that each basis pattern istruly representative of a phase, (2) excellent compositionspace connectivity of each phase concentration map, as ex-pected for equilibrium phase behavior, (3) systematic com-positional variation in the shift parameter λ, demonstrat-ing alloying within the phases, in particular phases 3, 5,and 6. While the AgileFD solution includes pattern shift-ing, the lack of adherence to the Gibbs phase rule has im-portant consequences on the solution that cannot be amelio-rated through modification of the H to impose Gibbs phaserule ex post facto, without a corresponding refinement ofW . The most prominent difference is highlighted in Fig-ure 5.2 where the second basis pattern of the AgileFD basispattern is quite different from that of the AgileFD-sparsity-gibbs solution. This AgileFD basis pattern contains a mix-ture of signals from other phases and is thus not able to bematched to a known structure. The researcher would thus beleft with the impression that this may be a newly discoveredphase and undergo the arduous task of hypothesizing newcrystal structures that could correspond to this basis pattern.The false-positive phase discovery from AgileFD is circum-vented in the AgileFD-sparsity-gibbs solution as the encod-
ing of Gibbs phase rule in the solver results in effective de-mixing of basis patterns such that all 6 primary phases areidentified. Indeed, only by imposing a priori constraints inAgileFD is a complete, meaningful solution produced.
6 ConclusionHigh-throughput materials discovery is revolutionizing theefficiency of materials science experiments through auto-mated characterization of thousands of materials in a sin-gle library. A major, critically-missing component of thehigh throughput materials discovery pipeline is the abil-ity to rapidly solve the phase map identification problem,which involves the determination of the underlying phasediagram of a family of materials from their compositionand structural characterization data. To address this substan-tial challenge, we developed Phase-Mapper, a comprehen-sive AI platform that tightly integrates XRD experimenta-tion, AI problem solving, and human intelligence for solv-ing the phase mapping problem. AI solvers in Phase-Mapperprovide, within minutes, high-quality solutions to the phasemapping problem, which can then be examined and fur-ther refined by materials scientists, interactively and in realtime. The de-mixing algorithm is a cornerstone of the Phase-Mapper platform, and previously-developed algorithms rou-tinely produce non-physical solutions. We have developeda novel solver, AgileFD, that features lightweight iterativeupdates of candidate solutions and a suite of novel adapta-tions to the multiplicative updating rules. In particular, wehave developed the ability to incorporate constraints thatcapture the physics of materials as well as human feedback,enabling functionalities well beyond traditional de-mixingtechniques and producing physically-meaningful solutions.We compare different solver variants with previously pro-posed methods in a large-scale experiment involving 20 syn-thetic systems, demonstrating the efficacy of imposing phys-ical constrains using AgileFD, while keeping fast solutiontimes. Phase-Mapper has been deployed in the Joint Cen-ter for Artificial Photosynthesis (JCAP) supported by theDepartment of Energy for materials scientists to solve awide variety of real-world phase diagrams, including thepreviously unsolved Nb-Mn-V oxide system, which is pro-vided here as a case study and an illustrative example ofthe benefits of encoding physical constraints. We believePhase-Mapper will lead to further developments in high-throughput materials discovery by providing rapid and crit-ical insights into the underlying phase behavior of new ma-terials.
AcknowledgementsThis material is supported by NSF awards CCF-1522054and CNS-0832782 (Expeditions), CNS-1059284 (Infras-tructure), and IIS-1344201 (INSPIRE); and ARO awardW911-NF-14-1-0498. Materials experiments are supportedthrough the Office of Science of the U.S. Department of En-ergy under Award No. de-sc0004993. Use of the StanfordSynchrotron Radiation Lightsource, SLAC National Accel-erator Laboratory, is supported by the U.S. Department ofEnergy, Office of Science, Office of Basic Energy Sciences
under Contract No. DE-AC02-76SF00515.
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