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Peiheng Zou, [Ryo Tamura](https://orcid.org/0000-0002-0349-358X), [Koji Tsuda](https://orcid.org/0000-0002-4288-1606)

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[Bayesian diversity control for batch-based phase diagram determination](https://mdr.nims.go.jp/datasets/17450f16-dc14-40a2-8faf-b050bd40c8a9)

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Bayesian diversity control for batch-based phase diagram determinationDigitalDiscoveryPAPEROpen Access Article. Published on 16 February 2026. Downloaded on 3/31/2026 10:34:52 PM.  This article is licensed under a Creative Commons Attribution-NonCommercial 3.0 Unported Licence.View Article OnlineView Journal  | View IssueBayesian diversitaGraduate School of Frontier Sciences, The UKashiwa, Chiba 277-8561, Japan. E-mail: tsbCenter for Basic Research on Materials, NTsukuba, Ibaraki 305-0047, JapancRIKEN Center for Advanced Intelligence Pr103-0027, JapanCite this: Digital Discovery, 2026, 5,1252Received 5th November 2025Accepted 13th February 2026DOI: 10.1039/d5dd00486arsc.li/digitaldiscovery1252 | Digital Discovery, 2026, 5, 125y control for batch-based phasediagram determinationPeiheng Zou,a Ryo Tamura abc and Koji Tsuda *abcMachine learning methods are increasingly used in experimental design in phase diagram determination.Some methods perform batch design, where multiple points are sampled from the design space. In thiscase, it is essential to diversify samples to avoid performing almost identical experiments, and control thediversity level appropriately. Manual diversity control is unintuitive and may require additional trial-and-error in prior to the experiments are started. We propose a Bayesian model called determinantal pointprocess for phase diagram construction (DPP-PDC) that can perform batch design and automaticdiversity control simultaneously. Central to this model is the uncertainty-weighted determinantal pointprocess that samples a set of points with high uncertainty under diversity control. Experiments with Cu–Mg–Zn ternary system demonstrate that DPP-PDC can actively control the sample diversity to achievehigh efficiency.1. IntroductionA phase diagram maps various phases of a material dependingon theomodynamic variables such as composition, temperatureand pressure.1 Numerous phase diagrams have been drawn foralloys and compounds2–4 and magnetic strutures.5–7 Deter-mining the phase diagram of a material is an essential step inmaterials development, but it requires a considerable numberof well-designed experiments by experts with changing experi-mental parameters, which is cost-intensive in terms of time andhuman effort.Some machine learning methods predict the entirety ofa phase diagram based on available data, without requiring anyexperiments.8–10 Meanwhile, active learning methods aim toguide experiments by recommending experimental parametersfor the next attempt.11–15 In formalization of active learning,a black-box function that maps a design space to an outcomespace is given. In phase diagram determination, the designspace is specied by the experimental parameters and theoutcome space is the set of all possible phase labels. The eval-uation of the black-box function at a point in the design spacecorresponds to phase measurement. The purpose of activelearning is to estimate the black-box function with as few eval-uations as possible. Given existing data, an active learningalgorithm recommends a point in the design space for nextevaluation. Intuitively, it is better to suggest the points close toniversity of Tokyo, 5-1-5 Kashiwa-no-ha,uda@k.u-tokyo.ac.jpational Institute for Materials Science,oject, 1-4-1 Nihonbashi, Chuo-ku, Tokyo2–1256phase boundaries to estimate the function quickly. An activelearning method, PDC,14 based on label propagation16 anduncertainty sampling,17 recommends the point of highest phaseuncertainty. PDC has been favorably tested both in bench-marks,14,18 and a real-world study identifying the phase diagramfor Zn–Sn–P lm deposition using molecular beam epitaxy.19 Inaddition, web application AIPHAD18 offers its active learningservice based on PDC.Active learning algorithms are classied into two types:single-probe and batch-based.20 PDC is a single-probe method,where the function evaluation is done one-by-one. Batch-basedmethods assume that evaluations are conducted at multiplepoints at once. Recently, the importance of batch-basedmethods is increasing due to the advent of self-driving labora-tories.21 Tamura et al.15 proposed several batch-based methods,but users need to determine a variable to adjust sample diver-sity manually in advance. Mancias et al.22 proposed to takea percentage of points with maximum uncertainty derived viaa Gaussian process and apply a k-medioid clustering method tosample a batch of points. In this case, the percentage deter-mines the diversity. Fig. 1 shows the importance of diversitycontrol schematically. Misspecication of the diversity levelwould lead to disastrous results.To eliminate the need for manual diversity control, wepropose a Bayesian approach23 called DPP-PDC based on theuncertainty-weighted determinantal point process (UwDPP).Given a set of points and the kernel matrix describing closenessamong them, the determinant of the kernel matrix representstheir diversity. In the k-determinantal point process (k-DPP),24a subset of size k is assigned a probablity proportional to thedeterminant. UwDPP modies k-DPP so that the points withhigher phase uncertainty are more likely to be chosen. UwDPP© 2026 The Author(s). Published by the Royal Society of Chemistryhttp://crossmark.crossref.org/dialog/?doi=10.1039/d5dd00486a&domain=pdf&date_stamp=2026-03-17http://orcid.org/0000-0002-0349-358Xhttp://orcid.org/0000-0002-4288-1606http://creativecommons.org/licenses/by-nc/3.0/http://creativecommons.org/licenses/by-nc/3.0/https://doi.org/10.1039/d5dd00486ahttps://pubs.rsc.org/en/journals/journal/DDhttps://pubs.rsc.org/en/journals/journal/DD?issueid=DD005003Fig. 1 Diversity control in phase diagram determination. (Left) When diversity is too low, the samples are concentrated to one point. (Middle)When diversity is properly controlled, they distribute close to phase boundaries. (Right) When diversity is too high, the samples are scattered allover the phase diagram.Fig. 2 Graphical model of DPP-PDC. Circular and square nodesrepresent random and non-random variables, respectively. The arrowsfrom X to Y represent that the generative model of Y depends on X.Paper Digital DiscoveryOpen Access Article. Published on 16 February 2026. Downloaded on 3/31/2026 10:34:52 PM.  This article is licensed under a Creative Commons Attribution-NonCommercial 3.0 Unported Licence.View Article Onlinehas a control variable to adjust how strongly diversity isimposed. A prior distribution is specied to the control variable,making DPP-PDC a Bayesian model. Markov chain Monte Carlosampling25 from the posterior distribution allows batchrecommendation with automatic diversity adjustment. NoteFig. 3 (a) Phase discovery curves. The x-axis corresponds to the number oRS stands for random sampling. (b) Evolution of control parameter s2.© 2026 The Author(s). Published by the Royal Society of Chemistrythat our method is applicable to any uncertainty measureincluding the one used by Mancias et al.22Using the phase diagram of Cu–Mg–Zn ternary system,26 wedemonstrate that DPP-PDC controls the sample diversity duringiterations appropriately to achieve high efficiency in activelearning.2. Method2.1. PDCLet us denote the set of candidate points in the design space byX = {x1, ., xN}. Usually, they are designated as grid points ina phase diagram. Let the set of possible phase labels be C.Assume that, for training data points T 4 [1, N], the phases{yi}i˛T are known. The task is to recommend a point not in T forour next evaluation. To infer the phases of the remainingpoints, the scikit-learn27 implementation of the label propaga-tion algorithm16 was used. A fully connected graph among X isconstructed rst, and each edge is weighted with an RBF kernelf samples and the y-axis shows the number of discovered phase labels.Digital Discovery, 2026, 5, 1252–1256 | 1253http://creativecommons.org/licenses/by-nc/3.0/http://creativecommons.org/licenses/by-nc/3.0/https://doi.org/10.1039/d5dd00486aDigital Discovery PaperOpen Access Article. Published on 16 February 2026. Downloaded on 3/31/2026 10:34:52 PM.  This article is licensed under a Creative Commons Attribution-NonCommercial 3.0 Unported Licence.View Article Onlinewith a lengthscale of 1/20. Based on the weighted graph, thelikelihood of point j belonging to phase c ˛ C is computed as fcj˛ [0, 1]. The uncertainty of point j is induced asuj ¼ 1�maxc˛Cfcj ; (1)and the point with maximum uncertainty jmax is chosen forrecommendation. As a result of the experiment, we obtain a newlabel yjmax. The point is added as T ) T W {jmax} and the aboveprocedure is repeated until the budget is met. In early itera-tions, it is likely that not all phase labels are included in {yi}i˛T.In that case, the uncertainty (1) is computed only with theexisting labels. As iterations go on, previously unseen phaselabels are discovered increasingly.2.2. Determinantal point processLet L is a kernel matrix among x1,., xN. Commonly, the kernelis determined as a Gaussian kernel, L(xi, xj)= exp(−g‖xi− xj‖2),where g determines its width. Determinantal point processeshave been used for sampling a diverse subset S from X invarious machine learning applications.24,28–30 The most funda-mental form of DPP is L-ensemble,PDPPðSÞ ¼ det LSdetðI þ LÞ ; (2)where LS is the submatrix of L restricted to S. For two points, it iseasy to understand that L-ensemble assigns high probability todistant pairs, becausedet LS = L(x1, x1)L(x2, x2) − L(x1, x2)2.In the two points are close, L(x1, x2) gets large, leading to smallPDPP. In batch recommendation, it is convenient to havea cardinality constraint. In such a case, one can use k-DPP,29Pk-DPPðSÞ ¼ det LSPS0˛X kdet LS0;where X k is the set of all size-k subsets of X. Kathuria et al.28found it useful to have a parameter to control the diversity andproposed k-DPP with mutual information kernel:Pk-DPPMIs ðSÞ ¼ detðI þ LS=s2ÞPS0˛X kdet�I þ LS0.s2� ; (3)where s is a control parameter. Let l1, ., ljSj denote theeigenvalues of LS. Then, the numerator of (3) is described asdet�I þ LS�s2� ¼YjSji¼1�1þ s�2li�:If s is small and two samples in S are identical, one of theeigenvalue li becomes zero and the probability of S is extremelysmall. In general, Pk-DPPMIs (S) is larger when the points arefarther from each other, and s−2 determines the strength of therepulsive force. When s approaches to innity, Pk-DPPMIs(S) converges to the uniform distribution, completelylosing the ability to impose diversity.1254 | Digital Discovery, 2026, 5, 1252–12562.3. DPP-PDCOur batch recommendation method, DPP-PDC, uses the samemachine learning model as PDC. Given the training data T, itprovides the phase uncertainty score U = {ui}i=1N for all points.Instead of choosing the point of maximum phase uncertainty,we need to pick up several diverse points for batch recom-mendation. To this aim, k-DPP with mutual information kernelis modied such that those with high uncertainty scores aremore likely to be chosen,PUwDPPs ðSÞ ¼Pk-DPPMIs ðSÞQi˛SuiPS0˛X kPk-DPPMIs�S0�Qi˛S0ui: (4)We call this distribution uncertainty-weighted DPP. To auto-mate the choice of parameter s, the prior distribution of s asa log-normal distribution,log�s2� � N ðm;uÞ;where hyperparameters are xed as follows: m = −4 and u = 4.The graphical model of this Bayesian model is described asFig. 2.The posterior distribution of this Bayesianmodel P(SjU, m, u)cannot be described analytically. However, sampling S and s ispossible with a Markov chain Monte Carlo (MCMC) method.23In MCMC, samples are perturbed randomly and the movementis either accepted or canceled according to a certain rule.Among a plethora of MCMC algorithms, we chose Non-U-Turnsampler31 implemented in pyMC.25 This algorithm is particu-lary useful, because the step size in perturbation is automati-cally derived. As for S, the perturbation is done by replacing oneelement randomly. The sampler starts from a random point andthe 1000-th sample is adopted as the nal solution of S and s.3. ResultsTo test DPP-PDC, we used the phase diagram of Cu–Mg–Znternary system.26 This alloy has applications in automotive andaerospace industries due to low density and exceptionalstrength. The diagram was calculated by CALPHAD, butconrmed to match available experimental measurements.26This phase diagram has three degrees of freedom, that is,fraction of Cu, fraction of Mg and the temperature. Thetemperature ranges from 500 K to 1500 K. Coexisting phases aregiven distinct labels (e.g., HCP-ZN + MG2ZN11). This resulted in71 phase labels in total. Candidate points are dened as the gridpoints, where 1% and 50 K intervals are adopted for the frac-tions and the temperature, respectively.For different batch sizes 5, 8, 10 and 12, DPP-PDC is applieduntil the number of total samples reach 300. In PDC, top-ksamples in terms of the uncertainty are taken as a batch. Fig. 3ashows the number of discovered phase labels against thenumber of samples (i.e., phase discovery curves). The curve anderror bar correspond to the average and standard deviation over10 runs, respectively. Fig. 4 shows the area under the phasediscovery curve for random sampling, PDC and DPP-PDC.Evaluations with respect to other performance measures are© 2026 The Author(s). Published by the Royal Society of Chemistryhttp://creativecommons.org/licenses/by-nc/3.0/http://creativecommons.org/licenses/by-nc/3.0/https://doi.org/10.1039/d5dd00486aFig. 4 Area under the phase discovery curve for random sampling(RS), PDC and DPP-PDC.Paper Digital DiscoveryOpen Access Article. Published on 16 February 2026. Downloaded on 3/31/2026 10:34:52 PM.  This article is licensed under a Creative Commons Attribution-NonCommercial 3.0 Unported Licence.View Article Onlineshown in Fig. S1 in the SI. As expected, DPP-PDC and PDCperformed signicantly better than random sampling. In allcases, DPP-PDC performed better than PDC, showing the meritof diversity control.Fig. 3b shows the evolution of control parameter s2 fordifferent batch sizes. Although s2 has certain variability amongiterations, we can observe the clear trend that a smaller value ofs2, therefore high diversity, is preferred as the number ofbatches increases. It matches our intuition that diversity mustbe imposed more strongly in large batch cases to avoidunwanted concentration of recommendations. This resultdemonstrates that DPP-PDC has an ability to control diversityappropriately and can at least avoid a catastrophic failurecaused by extremely small or large diversity.We tried different hyperparameter settings of DPP-PDC, butthe results showed little difference (Fig. S2 in the SI). It isbecause our prior distribution is set to enforce a minimallyweak constraint to allow adaption to data.We applied the batch sampling method by Mancias et al.,22where the cut-off parameter is set to 2.5% as instructed in thepaper. As shown in Fig. S3 in the SI, its performance was not asgood as ours. Batch optimization methods are known to besensitive to diversity parameter. Bayesian adaptation of the cut-off parameter to our dataset may have improved the perfor-mance, but it is out of scope of this paper.4. ConclusionIn this paper, we have shown how Bayesian modeling can beapplied to batch-based phase diagram determination. Thisapproach is more theoretically principled thanmanual diversitycontrol, and may nd various applications in self driving labo-ratories. Non-Bayesian heuristic methods do not have system-atic ways of parameter determination. Their parameters areoen determined by prior trial-and-error. When data acquisi-tion is costly as in self-driving laboratories, the cost of trial-and-error would exceed the computational cost of MCMC samplingnecessary in Bayesian methods.Experimental phase measurements may contain noise. Atphase boundaries, mislabeling due to noise is likely to occur.We used computational phase diagrams only in our experi-ments, and did not consider experimental noise. To apply DPP-© 2026 The Author(s). Published by the Royal Society of ChemistryPDC in experimental phase diagrams, some countermeasuresagainst noise may be necessary.One drawback of our approach is that the prior distributionof s2 is determined in an ad-hoc manner. Given a plenty ofphase diagram data, however, it could be determined by crossvalidation or related model selection methods. Anotherrestriction is that the sampling points are limited to the gridpoints. It may be problematic when a phase diagram of higherresolution is needed. It would be an interesting future work todevelop a Bayesian method for completely continuous settings.Author contributionsR. T. and K. T. conceived the idea and designed the research. P.Z. implemented the algorithm, and performed computationalexperiments. All authors wrote the manuscript.Conflicts of interestThe authors have no conicts to disclose.Data availabilityCode and datasets used in the paper are available at Githubhttps://github.com/tsudalab/DPP-PDC and Zenodo https://doi.org/10.5281/zenodo.18627155.Supplementary information (SI) is available. See DOI: https://doi.org/10.1039/d5dd00486a.AcknowledgementsThis work was supported by JSPS Kakenhi 25K01492, JST CRESTJPMJCR21O2 and ERATO JPMJER1903.References1 Y. A. Chang, S. Chen, F. Zhang, X. Yan, F. Xie, R. Schmid-Fetzer and W. A. Oates, Phase diagram calculation: past,present and future, Prog. Mater. Sci., 2004, 49, 313–345.2 K. Kennedy, T. Stefansky, G. Davy, V. F. Zackay andE. R. 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Published by the Royal Society of Chemistryhttps://doi.org/10.48550/arXiv.2507.22558http://creativecommons.org/licenses/by-nc/3.0/http://creativecommons.org/licenses/by-nc/3.0/https://doi.org/10.1039/d5dd00486a Bayesian diversity control for batch-based phase diagram determination Bayesian diversity control for batch-based phase diagram determination Bayesian diversity control for batch-based phase diagram determination Bayesian diversity control for batch-based phase diagram determination Bayesian diversity control for batch-based phase diagram determination Bayesian diversity control for batch-based phase diagram determination Bayesian diversity control for batch-based phase diagram determination Bayesian diversity control for batch-based phase diagram determination Bayesian diversity control for batch-based phase diagram determination Bayesian diversity control for batch-based phase diagram determination Bayesian diversity control for batch-based phase diagram determination Bayesian diversity control for batch-based phase diagram determination