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Akihiro Fujii, [Anh Khoa Augustin Lu](https://orcid.org/0000-0003-4702-0933), Koji Shimizu, [Satoshi Watanabe](https://orcid.org/0000-0002-8069-6938)

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[A straightforward gradient-based approach for designing superconductors with high critical temperature: exploiting domain knowledge                    <i>via</i>                    adaptive constraints](https://mdr.nims.go.jp/datasets/fd90e605-478b-4c7c-aad2-0306a05cbd39)

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A straightforward gradient-based approach for designing superconductors with high critical temperature: exploiting domain knowledge via adaptive constraintsDigitalDiscoveryPAPEROpen Access Article. Published on 29 October 2025. Downloaded on 2/4/2026 7:46:00 AM.  This article is licensed under a Creative Commons Attribution 3.0 Unported Licence.View Article OnlineView Journal  | View IssueA straightforwardaThe University of Tokyo, Department of MatBldg. IV, 7-3-1, Hongo, Bunkyo-ku, Tokyo 113t.u-tokyo.ac.jpbNational Institute for Materials Science (N0044, JapancNational Institute of Advanced IndustriaUmezono, Tsukuba, Ibaraki 305-8568, JapanCite this: Digital Discovery, 2025, 4,3662Received 5th June 2025Accepted 27th October 2025DOI: 10.1039/d5dd00250hrsc.li/digitaldiscovery3662 | Digital Discovery, 2025, 4, 36gradient-based approach fordesigning superconductors with high criticaltemperature: exploiting domain knowledge viaadaptive constraintsAkihiro Fujii, *a Anh Khoa Augustin Lu, ab Koji Shimizu cand Satoshi Watanabe aMaterials design aims to discover novel compounds with desired properties. However, prevailing strategiesface critical trade-offs. Conventional element-substitution approaches readily and adaptively incorporatevarious domain knowledge but remain confined to a narrow search space. In contrast, deep generativemodels efficiently explore vast compositional landscapes, yet they struggle to flexibly integrate domainknowledge. To address these trade-offs, we propose a gradient-based material design framework thatcombines these strengths, offering both efficiency and adaptability. In our method, chemicalcompositions are optimised to achieve target properties by using property prediction models and theirgradients. In order to seamlessly enforce diverse constraints—including those reflecting domain insightssuch as oxidation states, discretised compositional ratios, types of elements, and their abundance, weapply masks and employ a special loss function, namely the integer loss. Furthermore, we initialise theoptimisation using promising candidates from existing datasets, effectively guiding the search away fromunfavourable regions and thus helping to avoid poor solutions. Our approach demonstrates a moreefficient exploration of superconductor candidates, uncovering candidate materials with higher criticaltemperature than conventional element-substitution and generative models. Importantly, it couldpropose new compositions beyond those found in existing databases, including new hydridesuperconductors absent from the training dataset but which share compositional similarities withmaterials found in the literature. This synergy of domain knowledge and machine-learning-basedscalability provides a robust foundation for rapid, adaptive, and comprehensive materials design forsuperconductors and beyond.1 IntroductionMaterials design is crucial for technological innovation such asthe discovery of new superconductor materials. High-temperature superconductors (HTS) are especially promisingbecause they reduce cooling costs and enable higher magneticelds. They are also expected to be applied in fusion powergeneration, electric power cables, and superconducting maglevtrains.1,2Exploiting physical insights—such as selection of elementsbased on their oxidation states and the fact that materials withtoo many elements are impractical—can narrow this search,erials Engineering, Faculty of Engineering,-8656, Japan. E-mail: akihiro.fujii@cello.IMS), 1-1 Namiki, Tsukuba, Ibaraki 305-l Science and Technology (AIST), 1-1-162–3673making materials design more efficient. A traditional techniquein materials design is elemental substitution (i.e., doping).3–6 Inthis approach, one starts with a promising host material andpartially substitutes certain elements to tune the properties.Substituted elements are typically chosen based on physicalinsights—such as oxidation states—to ensure charge neutralityand other key constraints.Machine learning (ML) has become a widely used approachfor materials discovery, offering faster property predictions thanconventional Density Functional Theory (DFT) calculations andthus enabling high-throughput screening. In the context of HTSdevelopment, some studies7–15 have focused on training super-conducting transition temperature (Tc) prediction models usingthe SuperCon dataset,16 which comprises a large set of knownsuperconductors. Some studies17–19 combine ML-based Tcprediction with experimental tests and report the discovery ofnovel superconducting materials.Recently, deep generative models have gained prominencein materials design,20–23 including the quest to discover novel© 2025 The Author(s). Published by the Royal Society of Chemistryhttp://crossmark.crossref.org/dialog/?doi=10.1039/d5dd00250h&domain=pdf&date_stamp=2025-11-29http://orcid.org/0009-0002-2123-7958http://orcid.org/0000-0003-4702-0933http://orcid.org/0000-0001-5622-9582http://orcid.org/0000-0002-8069-6938http://creativecommons.org/licenses/by/3.0/http://creativecommons.org/licenses/by/3.0/https://doi.org/10.1039/d5dd00250hhttps://pubs.rsc.org/en/journals/journal/DDhttps://pubs.rsc.org/en/journals/journal/DD?issueid=DD004012Paper Digital DiscoveryOpen Access Article. Published on 29 October 2025. Downloaded on 2/4/2026 7:46:00 AM.  This article is licensed under a Creative Commons Attribution 3.0 Unported Licence.View Article Onlinesuperconductors.24,25 These models propose new compounds bylearning the statistical distribution of existing data, thusenabling the exploration of a vast chemical space. Severalstudies26,27 employ diffusionmodels28—a deep generative modelwidely used in the computer vision eld29—to generate super-conductor candidates. SuperDiff,27 a diffusion model forsuperconductors, generates candidate superconductors bygradually removing noise from a noisy composition. Moreover,SuperDiff can generate conditioned outputs based on referencecompounds using Iterative Latent Variable Renement (ILVR).30While conventional generative methods only explore materialswithin existing databases, SuperDiff can generate newmaterialsbased on promising reference compounds.Moreover, there are strategies that guide deep generativemodels toward desired properties, such as label-based condi-tional generation,31 Universal Guidance32 (UG), ClassierGuidance (CG)33 and Classier-Free Guidance (CFG).34 While CGand UG use a separate property predictor to steer the generationprocess, CFG does not require such a predictor. Although thelabel-based conditional generation and CFG have both beenextensively validated in image generation, their reliance onlabels within the dataset limits their exibility in materialsdesign. By contrast, CG can be conditioned on labels notpresent in the target dataset using models trained on otherdatasets. Xie et al.20 employ a strategy similar to CG, combininga diffusion model with a formation energy prediction model.Applying CG to Tc prediction models and superconductingFig. 1 Overview of Knowledge-Integrated Adaptive Gradient-based Optthe formation energy by optimising the input composition using twcomposition vectors, masks, and specialised loss functions, KIAGO enableSpecific elements can be fixed and excluded from the optimisation targwhich elements appear during the optimisation via masks. Here, we fixusing the mask; (2) restricting the maximum number of elements. We fia mask to keep only the most abundant ones up to a specified cutoff.elements never exceeds the chosen limit. In this figure, we select the threcomposition to those three elements. (3) Normalising the compositionalnormalised composition to a composition consisting of four atoms.© 2025 The Author(s). Published by the Royal Society of Chemistrymaterial generation models such as SuperDiff has the potentialto enable HTS design.A gradient-based method35–38 that uses prediction modelsand their gradients to optimise inputs has recently attractedattention. This method is similar to CG and UG but simpler, asit does not require training a generative model. Moreover, thismethod allows for more exible and adaptive conditionaloptimisation.38 While there is no study of applying this tech-nique to composition optimisation, it could be a promisingapproach for materials design.Despite these advances, signicant trade-offs remain.Elemental substitution can incorporate physical knowledge butmay limit exploration to a relatively narrow search space. Deepgenerative models can explore a broader chemical space effi-ciently, yet they struggle to exibly integrate physical knowl-edge—such as atomic valence constraints or convertingcompositional ratios to integers—in an adaptive manner. Onthe other hand, the gradient-based method has a risk of fallinginto poor solutions, though this method has the potential tointroduce various physical knowledge in an adaptive manner.In this paper, to address these issues, we adopt a gradient-based method and propose a straightforward materials-designmethod called Knowledge-Integrated Adaptive Gradient-basedOptimisation (KIAGO). This framework combines the adaptiveapplication of domain knowledge with computational efficiencyto directly optimise chemical compositions (Fig. 1). KIAGO doesnot require training a deep generative model, making it morestraightforward to implement. Specically, we adopt twoimisation (KIAGO). KIAGO simultaneously maximises Tc and minimiseso pretrained models and their gradients. Through the use of fixeds flexible control of the composition in three ways: (1) element control.et to perform conditional optimisation. KIAGO is also able to controlthe composition of barium and exclude helium from the optimisationrst rank elements by their abundance in the composition and createAll other elements are set to zero, ensuring that the total number ofemost abundant elements to build a mask, which then restricts the finalratios to small integers. Here, we use the loss function Lint4 to guide theDigital Discovery, 2025, 4, 3662–3673 | 3663http://creativecommons.org/licenses/by/3.0/http://creativecommons.org/licenses/by/3.0/https://doi.org/10.1039/d5dd00250hDigital Discovery PaperOpen Access Article. Published on 29 October 2025. Downloaded on 2/4/2026 7:46:00 AM.  This article is licensed under a Creative Commons Attribution 3.0 Unported Licence.View Article Onlineproperty prediction models—one for Tc and another forformation energy—to maximise Tc while enhancing stability,thereby proposing realistic materials. Unlike CFG and label-based conditional generation, KIAGO can optimise formationenergy (which is not included in the SuperCon dataset) by usinga separately trained formation energy prediction model. Addi-tionally, by conducting an intensive search around promisingmaterials, KIAGO mitigates the risk of being trapped at poorresults. Moreover, by applying masks and a specialised lossfunction to enforce integer values, we can effectively embedphysical insights (e.g., ensuring the retention of specicelements, oxidation states, the number of elements, or integercompositional ratios), thus providing a versatile framework thataccommodates diverse constraints in adaptive manners.To validate the effectiveness of KIAGO, we performedexperiments to propose promising HTS. Our approach signi-cantly outperformed both generative models (SuperDiff andSuperDiff with CG) and conventional elemental substitutiontechniques in proposing high-Tc candidates efficiently. Inparticular, we found that SuperDiff with CG tended to generatematerials with lower Tc values because the Tc distribution in theoriginal data constrained them. In contrast, our methodproposed high-Tc candidates without being limited by theoriginal distribution. Additionally, KIAGO could keep some partof the composition xed while optimising others, relevantlyreplace elements according to their oxidation states, andmaintain charge neutrality perfectly. Additionally, KIAGOproposed candidate compositions that shared the sameelements as hydride superconductors reported in other litera-ture despite their absence from the SuperCon dataset. Theseresults highlight its potential for discovering novel materials.2 KIAGO2.1 OverviewKnowledge-Integrated Adaptive Gradient-Based Optimisation(KIAGO) is a gradient-based method that uses pre-trainedmodels and their gradients to directly optimise the inputrepresentation—in this case, the normalised compositionalvector of candidate materials. Rather than merely searching forcompositions that yield favourable properties, KIAGO intro-duces three key strategies to enhance material quality andprovide ne-grained control: (1) initialisation from promisingmaterials to mitigate the risk of being trapped at poor results;(2) masking to control elemental types; (3) special loss functionsfor conversion to integers and atomic-count constraints.2.2 Gradient-based methodA gradient-based method can adopt any predictive model,provided the chain rule of differentiation is valid from input tooutput. To propose superconducting materials with high-Tc, weemploy a Tc predictor fTc. We also use a formation-energypredictor fEfto propose compounds that are both high-Tc andthermodynamically feasible. We introduce a hyperparametera and dene the loss L as3664 | Digital Discovery, 2025, 4, 3662–3673L = −fTc(x̂) + afEf(x̂) (1)x̂* ¼ argminx̂L: (2)Here, x̂ ˛ [0, 1]Nelem is a compositional vector spanning Nelemelements. Minimising L aims to increase Tc while lowering theformation energy. However, simple minimisation poses severalissues: (1) it may converge to poor solutions, (2) it lacks controlover the number and type of elements, and (3) it does not ensureinteger ratios in the nal composition.2.3 Initialisation based on promising materialsTo avoid converging to poor solutions, we adopt a strategy ofstarting the optimisation from various initial states includingthose corresponding to known promising materials. We canreduce this risk by focusing on the areas of existing high-performance compounds. Such a strategy goes beyonddoping-like approaches that only alter part of an existingmaterial, enabling a broader range of materials to be explored.Specically, we perturb known superconductors by substitutingelements randomly and adding new elements to the composi-tion. This technique effectively explores the local neighbour-hood of promising materials.2.4 Controlling the types of elements presentWe next control which elements appear in the composition bycombining a xed composition vector and a mask (Fig. 1(1)).First, we split x̂ into a xed portion xconst and an optimisableportion x̂opt:x̂opt˛ℝNelem ; xconst˛½0; 1�Nelem ;Xixconsti\1 (3)x̂ ¼ xconst þ s�x̂opt� 1�Xixconsti!: (4)Here, s is a normalisation function that ensures each element isnon-negative and the total sum is 1. A possible approach was touse the somax function. However, to emphasize elements thatremain unused, we instead chose to use normalisation aerapplying Rectied Linear Unit (ReLU).sðxÞ ¼ ReLUðxÞPReLUðxÞ (5)Because xconst remains unchanged, its specied compositionremains xed during optimisation. We further introducea mask Melem(Melem ˛ {0, 1}Nelem) to select the allowableelements. Concretely, we setx̂opt = x̂base*Melem, (6)where x̂baseðx̂base˛ℝNelemÞ is a trainable parameter, and theasterisk (*) denotes element-wise multiplication. This maskenforces strict control over which elements can be used, thusguiding the optimisation toward compositions that meet spec-ied domain constraints.© 2025 The Author(s). Published by the Royal Society of Chemistryhttp://creativecommons.org/licenses/by/3.0/http://creativecommons.org/licenses/by/3.0/https://doi.org/10.1039/d5dd00250hPaper Digital DiscoveryOpen Access Article. Published on 29 October 2025. Downloaded on 2/4/2026 7:46:00 AM.  This article is licensed under a Creative Commons Attribution 3.0 Unported Licence.View Article Online2.5 Controlling the number of element types in thecompositionTo achieve realistic composition, we use a mask to limit howmany elements can appear in the composition (Fig. 1(2)). To dothis, we sort the compositional values in ascending order andcreate a mask Mmaxelem(Mmaxelem ˛ {0, 1}Nelem), which sets to zero anyelement index beyond the allowed maximum elements nmaxelem.PMmaxelem = nmaxelem (7)x̂0opt ¼ x̂opt*Mmaxelem; (8)2.6 Integer lossWe construct a loss function that guides the composition intointeger-compatible values during optimisation (Fig. 1(3)). Suchvalues {cnNunit} are those for which the product of the normalisedcomposition and the unit cell size Nunit becomes an integerratio. For instance, if Nunit = 4, then the feasible set {cn4} is{0.00, 0.25, 0.50, 0.75, 1.00}.The integer loss measures how far each compositional ratioof element i(x̂i) is from its nearest value in {cnNunit}.{cnNunit} = {n/Nunit}n=0,1,.,Nunitx̂i ˛ {x̂H, x̂He, x̂Li, ., x̂Nelem} (9)LintNunitðx̂Þ ¼XNelemi¼1minn��x̂i � cnN�� (10)As an example, Fig. 2 shows the result of applying Lint4 toCa0.23Sr0.27O0.50. We assume Nunit = 4 and guide the composi-tions toward the nearest values in {0.00, 0.25, 0.5, 0.75, 1.00}.Fig. 2 An overview of the integer loss Lint4 under the assumption thateach unit cell contains four atoms. The numbers shown inside thedashed box represent all possible different combinations between theinteger-compatible set {cn4} and the composition values. The total lossis obtained by selecting the minimum among these combinations foreach element (indicated by the black underline) and summing them.© 2025 The Author(s). Published by the Royal Society of ChemistryBecause it is difficult to x Nunit in advance, we evaluatemultiple candidates for Nunit and choose the one that yields thesmallest loss. Concretely, we dene Linteger as follows:Linteger;fNunitgðx̂Þ ¼ minN˛fNunitgLintNðx̂Þ: (11)This exible approach selects a suitable integer grid evenwhen the optimal cell size is unknown.2.7 Optimisation procedureKIAGO divides its optimisation into two stages. First, as inSections 2.3 and 2.4, we construct an initial x̂base and controlwhich elements appear while iteratively minimising thefollowing loss L1st. This process yields x̂1st* .x̂ ¼ xconst þ sðx̂base*MelemÞ 1�Xixconsti!(12)L1st = −fTc(x̂) + afEf(x̂) (13)x̂1st* ¼ argminx̂baseL1st (14)Next, we introduce the conversion to integers anda maximum-atom constraint. We use x̂1st* to build the maskMmaxelem, then iteratively minimise the loss L2nd.x̂0 ðx̂baseÞ ¼ xconst þ s�x̂base*Mmaxelem*Melem� 1�Xixconsti!(15)L2nd = −fTc(x̂0) + afEf(x̂0) + bLinteger,{Nunit}(x̂0) (16)x̂2nd* ¼ argminx̂baseL2nd (17)Here, b is a hyperparameter. We take x̂0(x̂2nd* ) as the nalsolution.3 Results3.1 Implementation detailsWe used PyTorch39 to implement KIAGO. KIAGO optimisesa total of 4096 candidate compositions across 1000 steps usingAdam optimiser.40 We rst perform 500 steps of optimisationusing eqn (12)–(14), followed by another 500 steps using eqn(15)–(17) with a = 4 (selected based on tuning) and b = 1.To predict the superconducting transition temperature (Tc),we employ a ResNet18 model41 trained on normalised compo-sitions of SuperCon and Crystallography Open Database(COD).42–51 Each composition is represented by a periodic table-based feature map, which has four channels corresponding tothe s, p, d, and f orbitals.12 SuperCon comprises more than 26000 composition–Tc pairs and is widely used for Tc prediction.Although SuperCon lacks explicit structural information, it isused to propose novel superconductor candidates, some ofwhich are later veried experimentally.17–19 We also use COD asDigital Discovery, 2025, 4, 3662–3673 | 3665http://creativecommons.org/licenses/by/3.0/http://creativecommons.org/licenses/by/3.0/https://doi.org/10.1039/d5dd00250hFig. 3 Comparison of optimisation results under different initialisationmethods. Both approaches employ Adam optimiser40 with a learningrate of 0.001. (Left) Initialisation by adding noise to an existingsuperconductor (LaNiAsO). (Right) Random initialisation, in whichseven elements are chosen arbitrarily and assigned random compo-Digital Discovery PaperOpen Access Article. Published on 29 October 2025. Downloaded on 2/4/2026 7:46:00 AM.  This article is licensed under a Creative Commons Attribution 3.0 Unported Licence.View Article Onlinea source of non-superconductors to regularize training andreduce false positives.For each element, we set ags at the positions of its row andcolumn on the periodic table, as well as at its relevant orbitalchannels. We then multiply these element-level feature maps bythe respective compositional ratios to create the nal inputrepresentation. Further details are available in Section S.1. Forformation energy, we used ElemNet,52 which is originallyimplemented in TensorFlow 1.x53 and we re-implemented itin PyTorch. Since ElemNet only covers elements up toatomic number 86, we apply a mask to exclude elementsbeyond that range. Additional technical specics are given inSection S.2.We compared KIAGO against two baselines: a conventionalelemental-substitution (C-ES) approach and SuperDiff, a diffu-sion-based generative model. The C-ES method randomlyreplaces some elements with others of identical oxidationstates. For SuperDiff, we used the official implementation andtrained on the same data for our Tc prediction model, butwithout normalising compositions. Following the official code,we conducted 1000 diffusion steps. According to the originalSuperDiff, we conditioned generations of compositions onexisting superconductors using Iterative Latent VariableRenement (ILVR). We applied scale factors of 1, 2, 4, and 6 toyield a total of 4096 samples. We also incorporated ClassierGuidance (CG) using Tc prediction model and ElemNet intoSuperDiff to compare it directly with KIAGO. Normally, theclassier used for CGmust be trained on data with noise, whichwould make Universal Guidance (UG) the better choice for off-the-shelf models. However, we found that UG did not workwell and CG still improved Tc using models without noise-augmented training. Therefore, we decided to use CG.Although neither model is strictly a classier, we refer to thisapproach as CG for convenience. Note that this is not proposedin the original paper.27 At each inference step, we used eqn (13)with a = 4 for gradient guidance, and we tuned the guidanceweight from 1 × 10−7 to 1.0, ultimately selecting 1 × 10−3.Additional details of SuperDiff are provided in Section S.3.Aer generating candidate compositions, we applied amulti-step screening procedure to ensure realistic materials. First, weused SMACT54 to lter compositions with charge neutrality andelectronegativity balance, following Yuan et al.27 Next, weselected only those with formation energies (predicted by Ele-mNet) less than zero. We also removed compositions contain-ing ten or more elements since the preprocessed SuperCon datahave at most nine. Finally, we evaluated Tc values using thesame ResNet18 predictor used in KIAGO and SuperDiff with CG.To assess thermodynamic stability of the proposed mate-rials, we use the energy above the convex hull per atom (DEhull).Our method proposes compositions rather than crystal struc-tures, so validatingDEhull with DFT total energies is not feasible.Instead, we estimate DEhull from formation energies predictedby ElemNet. Concretely, we predict formation energies forcompositions from the Materials Project55 and the AlexandriaMaterials Database,56–59 and compute DEhull by building phasediagrams with pymatgen.603666 | Digital Discovery, 2025, 4, 3662–36733.2 Mitigating the risk of convergence to non-promisingsolutionsIn this section, we investigated whether our initialisationscheme could mitigate the risk of converging to non-promisinglocal minima in the rst stage (eqn (12)–(14)). Specically, weaimed to determine whether our method can produce morepromising local optima than a purely random initialisation. Inour method, we began with the superconductor LaNiAsO fromthe SuperCon dataset. With a probability of 0.22, we replacedelements of its composition with different elements chosenaccording to their occurrence frequencies in SuperCon. We thenselected random elements with random compositional ratios(from 0.0 to 0.3) for those elements, normalised the resultingcomposition, and used it as the initialisation. By contrast, therandom initialisation selects four elements, to match thenumber of atoms in LaNiAsO, uniformly at random and assignsthem random compositional values.Fig. 3 shows the optimisation results. Our initialisationscheme yields higher Tc values than random initialisation.Although our method can still become trapped in local optima,it proposes more promising solutions than the randomapproach. Hence, our approach partially mitigates the inherentchallenge of local minima in gradient-based methods. SeeSection S.4 for details on the variability across different randomseeds.3.3 Converting compositional ratios to integers via loss-based approachIn this section, we compared the integer conversion methodbased on a loss function Linteger,{Nunit} with a rule-based integerconversion method. We aimed to determine which approachreduced the drop in Tc in the second stage (eqn (15)–(17)). In theloss-based method, we follow eqn (16) to maximise Tc whileminimising Ef. Specically, we guided the composition towardan integer representation by selecting an optimal total numberof atoms from a set of integers smaller than the speciedsitional values.© 2025 The Author(s). Published by the Royal Society of Chemistryhttp://creativecommons.org/licenses/by/3.0/http://creativecommons.org/licenses/by/3.0/https://doi.org/10.1039/d5dd00250hTable 2 Differences in Tc (K) between the average Tc of the highesttop-30 proposed superconductors and the base superconductors inthe experiments of proposing superconductors based on existingones. SD, SD w/ CG, and C-ES denote SuperDiff, SuperDiff withclassifier guidance, and conventional elemental substitution, respec-tively. After screening 4096 samples, the Tc predictionmodel was usedto calculate Tc. ‘N/A’ indicates that none of the samples passed thescreeningBase materials from SuperConKIAGO SD SD w/ CG C-ESTop-30 Top-30 Top-30 Top-30DTc (K) DTc (K) DTc (K) DTc (K)LaNiAsO 104.39 −1.42 −0.47 13.54SrFe1.88Ni0.12As2 97.91 26.76 4.53 21.88Sr4V2Fe2As2O6 97.49 −13.77 −13.66 −5.45LaPt2B2C 86.73 −5.01 −4.66 6.90HgBa2Ca2Cu3O8 17.07 −29.43 −1.18 −12.11CeBiS2O 92.12 N/A −0.31 2.04Bi2Sr2CuO6 129.63 18.34 26.76 43.78TlSr2CaCu2O7 76.32 5.21 12.42 18.60Table 1 Comparison between the rule-based approach andLinteger,{Nunit} for converting compositional ratios to integers. The tableshows the average change in Tc before and after conversion to inte-gers under certain maximum numbers of atoms, based on a total of 61440 samples derived from 15 different superconducting materialsMax. num. atoms Linteger,{Nunit} (K) Rule-based (K)15 −3.59 −7.6520 −0.81 −4.3025 −0.93 −2.2350 −0.43 −1.38100 −0.13 −0.31Paper Digital DiscoveryOpen Access Article. Published on 29 October 2025. Downloaded on 2/4/2026 7:46:00 AM.  This article is licensed under a Creative Commons Attribution 3.0 Unported Licence.View Article Onlinemaximum atom count. We optimised them for 500 steps forinteger conversion. By contrast, the rule-based approachmultiplies the normalised composition by the specied numberof atoms and then rounds each value to the nearest integer.Table 1 shows the results comparing the rule-basedapproach and Linteger,{Nunit}. Because it remains closer to thepre-conversion to integers composition, rounding with a largertotal number of atoms is generally advantageous. Although therule-based method fully exploits this by always rounding at themaximum atom count, Linteger,{Nunit} does not always do so, yet itstill performs better.Table 3 Samples of proposed superconductorsMethod SamplesKIAGO Ca4Co3Sr3W3F2As8 (115.65 K)Ca5Cu4Sr5O11 (127.16 K)MgCa4Cu4Ba3TlO10 (142.90 K)Cu6Sr3Pt3B5O7 (87.52 K)SD Ni0.99LaO1.01As0.99 (3.65 K)Ni0.82Ge0.25La0.97C1.59As0.44Se0.14 (1.23 K)Ca0.28Cu1.39Sr2.04Pb0.91Bi1.15O7.03 (76.03 K)Ca1.84Sc0.17Cu2.91Ba2.05HgO8.05 (127.46 K)SD w/ CG CaCu2.02Sr1.69Y0.41Tl1.04O6.95 (87.78 K)Co0.3Ni0.69La0.83Ce0.16O0.94As0.97 (4.85 K)La0.6Ce0.43Nd0.15Bi1.03O0.98S2.01 (2.54 K)Ni0.9Ge0.11La0.96C0.29O0.87As0.8 (5.96 K)C-ES La4Bi4O4S8 (4.61 K)CaCu2BaTlCO7 (86.78 K)V2Sr4YHfO6As2 (36.65 K)La2Hf2Ir2B4C2 (18.46 K)3.4 Generating superconductors with higher Tc based onexisting onesIn this section, we investigated whether our method couldpropose superconductors with higher Tc values based on knownsuperconductors as initial candidates. For KIAGO, we start withexisting superconductors and introduce noise to the composi-tions described in Section 3.2. We used Adam optimiser withlearning rate of 0.03 for KIAGO. We set {Nunit} = {1, 2, ., 25}and nmaxelem = 10. SuperDiff conditions on existing superconduc-tors via Iterative Latent Variable Renement (ILVR). Theconventional elemental-substitution (C-ES) approach randomlyreplaces a subset of elements with others sharing the sameoxidation state. For each base material of superconductor, weselect copper-based, iron-based, and other superconductorsthat pass charge-neutrality and electronegativity screening bySMACT, then randomly choose from these sets as basematerials.Table 2 presents the differences in predicted Tc between thegenerated superconductors and their base materials. KIAGOachieves the most efficient exploration of higher Tc valuescompared to other methods. By contrast, there are experimentswhere SuperDiff does not yield any valid materials passing allscreenings. This limitation may stem from the fact that manyentries in SuperCon do not pass charge-neutrality and electro-negativity checks; hence, the model struggles to generate validcompositions. The C-ES method also fails to propose suffi-ciently high Tc compounds, likely because its rule-basedapproach cannot fully explore the vast compositional space.In contrast, KIAGO proposes many materials showingsubstantial Tc increases. For completeness, Section S.6 includesall screening-pass rates.© 2025 The Author(s). Published by the Royal Society of ChemistryTable 3 describes example compositions. Both KIAGO and C-ES yield integer-total compositions, allowing straightforwardinduction of possible crystal structures. However, SuperDiff andSuperDiff with CG frequently produce non-integer totals,making immediate structural analysis more challenging. Forseveral candidates proposed by KIAGO, we computed theconvex-hull distance DEhull using ElemNet. Ca5Cu4Sr5O11(127.16 K) and MgCa4Cu4Ba3TlO10 (142.90 K) showed DEhull <0.06 (eV per atom), suggesting possible thermodynamicstability.Interestingly, SuperDiff with CG does not necessarilygenerate higher-Tc compounds than SuperDiff alone. Table 4shows how Tc changes in guidance and denoising underDigital Discovery, 2025, 4, 3662–3673 | 3667http://creativecommons.org/licenses/by/3.0/http://creativecommons.org/licenses/by/3.0/https://doi.org/10.1039/d5dd00250hTable 4 Total changes in Tc resulting from guidance, ILVR, and denoising during the 1000 steps in SuperDiffw/ CG. “Guide weight” denotes theweight for guidance. “Denoise DTc”, “ILVR DTc” and “Guide DTc” represent the cumulative change in Tc per step due to denoising, ILVR, or theguidance. “Sum” is the total of these values. “Screening ratio” denotes the ratio of the number of screened samples to the total number of samplesGuide weight w Denoise DTc (K) ILVR DTc (K) Guide DTc (K) Sum (K) Screening ratio— 84.8 37.4 — 122.2 0.131.0 × 10−5 80.2 43.0 0.0 123.2 0.091.0 × 10−4 75.9 46.9 0.4 123.1 0.101.0 × 10−3 79.2 39.8 3.2 122.3 0.081.0 × 10−2 77.1 13.2 33.8 124.1 0.111.0 × 10−1 42.5 −197.1 279.3 124.7 0.121.0 −63.5 −1409.6 1611.0 137.9 0.001.0 × 101 −101.7 −3945.1 4190.5 143.7 0.001.0 × 102 −42.2 −3746.6 3925.2 136.4 0.00Digital Discovery PaperOpen Access Article. Published on 29 October 2025. Downloaded on 2/4/2026 7:46:00 AM.  This article is licensed under a Creative Commons Attribution 3.0 Unported Licence.View Article Onlinedifferent weights for guidance. During generation, high guid-ance weights raise Tc in the guidance step but then revert it inthe ILVR and denoising step. We attribute this to the Tc distri-bution in SuperCon, where low- or moderate-Tc compoundsdominate (median: 12.5 K). As a result, extremely high Tc valuesare treated as noise, prompting the model to restore them tomore typical levels. Furthermore, larger weight for guidancecause a stronger mismatch with the training distribution,reducing the fraction of generated compositions that containfewer than ten elements. This interplay of denoising andguidance likely hampers SuperDiff with CG's ability to reachstable, high-Tc solutions. For the results without ILVR, pleaserefer to Section S.3.3.5 Elemental substitutionIn this section, we implemented elemental substitution toimprove Tc. Here we replaced one metal element based on itsoxidation state while retaining the rest of the composition.Specically, we chose a single metal element from an existingsuperconductor and kept the remaining composition xed.We implemented this approach in KIAGO by treating thepreserved composition as a xed vector xconst. We thenrandomly initialise the substituting element and apply a maskbased on its oxidation state. For example, when substitutingTable 5 Success rate of elemental substitution in proposed materials. Wfollowing criteria: (1) the preserved composition remains within 1% of itelement (or elements) stays within 1% of the original substituted metal'sBase materials from SuperCon Substitute targetCeFeAsF0.2O0.8 Ce3+LaFeAsO La3+SrFe2As2 Sr2+Bi2CaSr2Cu2O8 Bi3+CeNiC2 Ce4+LaNiC2 La3+MgCoNi3 Co2+RuSr2GdCu2O8 Sr2+RuSr2YCu2O8 Y3+Y2Fe3Si5 Y3+YIrSi Y3+3668 | Digital Discovery, 2025, 4, 3662–3673Y3+, we only allow elements having a +3 oxidation state, such asgallium or aluminum. To achieve this, we used a mask that hasone value on elements having a +3 oxidation state and set allothers to zero. To simplify evaluation, we excluded thepreserved elements from the mask. We use Adam optimiserwith learning rate of 0.03 for KIAGO. Note that we did notperform conversion to integers on the substituting element, sowe set b = 0. By contrast, SuperDiff does not explicitly supportelemental substitution, so we approximated it by conditioningthe generation process with ILVR.In addition to the screening described in Section 3.1, weassessed whether the intended elemental substitution wascorrectly carried out. First, we checked whether the preservedcomposition remains within 1% of its original ratio. Second, wechecked that the total composition of the newly substitutedelement (or elements) stays within 1% of the originalsubstituted metal's ratio. To simplify evaluation, we excludedthe preserved elements from the substituted element candi-dates. We then evaluated the Tc of compositions that pass boththis substitution check and the previous screening.Tables 5–7 summarize the probability of correct elementevaluation, the charge-neutrality evaluation, and the resultingTc values, respectively. Notably, KIAGO and C-ES achieve 100%correct substitutions (Table 5), indicating that these methodse defined a successful elemental substitution as satisfying both of thes original ratio, and (2) the total composition of the newly substitutedratioKIAGO SD SD w/ CG C-ES1.00 0.00 0.00 1.001.00 0.01 0.00 1.001.00 0.01 0.00 1.001.00 0.23 0.20 1.001.00 0.00 0.00 1.001.00 0.01 0.01 1.001.00 0.00 0.00 1.001.00 0.03 0.03 1.001.00 0.03 0.03 1.001.00 0.00 0.00 1.001.00 0.00 0.00 1.00© 2025 The Author(s). Published by the Royal Society of Chemistryhttp://creativecommons.org/licenses/by/3.0/http://creativecommons.org/licenses/by/3.0/https://doi.org/10.1039/d5dd00250hTable 6 Success rate with respect to charge neutrality in proposed materials resulting from elemental substitutionBase materials from SuperCon Substitute target KIAGO SD SD w/ CG C-ESCeFeAsF0.2O0.8 Ce3+ 1.00 0.03 0.02 1.00LaFeAsO La3+ 1.00 0.02 0.02 1.00SrFe2As2 Sr2+ 1.00 0.03 0.05 1.00Bi2CaSr2Cu2O8 Bi3+ 1.00 0.01 0.01 1.00CeNiC2 Ce4+ 1.00 0.07 0.06 1.00LaNiC2 La3+ 1.00 0.03 0.04 1.00MgCoNi3 Co2+ 1.00 0.71 0.63 1.00RuSr2GdCu2O8 Sr2+ 1.00 0.05 0.06 1.00RuSr2YCu2O8 Y3+ 1.00 0.08 0.07 1.00Y2Fe3Si5 Y3+ 1.00 0.18 0.26 1.00YIrSi Y3+ 1.00 0.28 0.48 1.00Table 7 Differences in Tc (K) between the average Tc of the highest top-30 proposed superconductors and the base superconductors inexperiments of elemental substitution. After screening 4096 samples, the Tc prediction model was used to calculate Tc. ‘N/A’ indicates that noneof samples passed the screeningBase materials from SuperCon Substitute targetKIAGO SD SD w/ CG C-ESTop-30 Top-30 Top-30 Top-30DTc (K) DTc (K) DTc (K) DTc (K)CeFeAsF0.2O0.8 Ce3+ 14.17 N/A N/A 9.96LaFeAsO La3+ 33.88 1.24 N/A 31.10SrFe2As2 Sr2+ 31.54 N/A N/A 19.25Bi2CaSr2Cu2O8 Bi3+ 18.72 0.15 −1.11 6.10CeNiC2 Ce4+ 13.10 N/A −0.02 4.33LaNiC2 La3+ 10.53 N/A N/A 4.90MgCoNi3 Co2+ 31.66 N/A 0.13 7.23RuSr2GdCu2O8 Sr2+ −3.38 −0.23 −0.16 5.64RuSr2YCu2O8 Y3+ 45.96 −0.89 1.53 37.74Y2Fe3Si5 Y3+ 5.57 N/A N/A 1.50YIrSi Y3+ 6.09 3.42 N/A 3.86Paper Digital DiscoveryOpen Access Article. Published on 29 October 2025. Downloaded on 2/4/2026 7:46:00 AM.  This article is licensed under a Creative Commons Attribution 3.0 Unported Licence.View Article Onlineincorporate domain knowledge effectively. Consequently, asshown in Table 6, their proposedmaterials always satisfy chargeneutrality. Moreover, KIAGO demonstrates high search effi-ciency, yielding the best results in most experiments (Table 7).SuperDiff, however, cannot reliably perform elemental substi-tution, indicating that generative models like SuperDiff are notwell suited when strict domain knowledge must be enforced.Next, we compare the highest-Tc compounds in the Super-Con dataset that have undergone the same elemental substi-tution with the compounds proposed by KIAGO. Table 8 showsthat, in most element-substitution experiments, KIAGOproposes materials with higher Tc than any element-substitutedmaterials in the SuperCon dataset. This result highlights thepotential of our approach to surpass known substitution strat-egies and discover more promising superconductors.For several candidates proposed by KIAGO, we computed theconvex-hull distance DEhull using ElemNet. Element-substitutedderivatives of Y2Fe3Si5 and CeFeAsF0.2O0.8—namely Pr0.3995-Gd0.3859Dy0.3634Ho0.361Hf0.2428Sm0.2474Fe3Si5 (Tc = 5.3 K) andAg0.1752Sm0.6868Tb0.138FeAsF0.2O0.8 (Tc = 51.66 K)—exhibitconvex-hull distances of DEhull = 0.02 and 0.00 eV per atom,respectively, indicating potential thermodynamic stability.© 2025 The Author(s). Published by the Royal Society of ChemistryIn this section, we limit our discussion to single-elementsubstitution. However, our method can also support multi-element substitution. For example, in Ti2O4, two Ti4+ atomsand one O2− atom contribute a total charge of +6. This can bereplaced by two X3+ atoms, resulting in a composition like X2O3.Here, X denotes any element with a +3 oxidation state. Suchsubstitutions are feasible as long as the total charge ispreserved, and our oxidation-state-based masking mechanismcan accommodate them.3.6 Proposing novel hydride superconductorsIn this section, we focused on proposing novel hydride super-conductors (HSC). Many known HSC are binary or ternarysystems containing hydrogen and just one or two otherelements, with hydrogen comprising a large fraction of thecomposition. Thus, we constrained KIAGO to compositions thathave at least 40% hydrogen to expand the space around existingmaterials, form binary or ternary compounds, and possessa total atom count of 15 or fewer. HSC are oen tested underhigh pressure, where Pauling's electronegativity rules may nothold. For instance, LaH10 has been experimentally conrmedbut fails SMACT-based screening for electronegativity andDigital Discovery, 2025, 4, 3662–3673 | 3669http://creativecommons.org/licenses/by/3.0/http://creativecommons.org/licenses/by/3.0/https://doi.org/10.1039/d5dd00250hTable 8 Comparison between materials proposed by KIAGO and the highest-Tc compounds from the SuperCon dataset under the sameelement-substitution conditions. All Tc values are prediction values by our Tc prediction model. Blue elements indicate the substituted elementsBase materials from SuperCon Substitute target Proposed samples Best in SuperConCeFeAsF0.2O0.8 (39.8 K) Ce3+LaFeAsO (13.4 K) La3+SrFe2As2 (17.3 K) Sr2+Bi2CaSr2Cu2O8 (81.0 K) Bi3+CeNiC2 (3.0 K) Ce4+LaNiC2 (2.4 K) La3+MgCoNi3 (7.4 K) Co2+RuSr2GdCu2O8 (33.7 K) Sr2+RuSr2YCu2O8 (34.4 K) Y3+Y2Fe3Si5 (2.3 K) Y3+YIrSi (2.8 K) Y3+Table 9 Average Tc for the top-5 proposed superconductorsBase materialsfrom SuperConKIAGO SD SD w/ CG C-ESTop-5 Top-5 Top-5 Top-5Tc (K) Tc (K) Tc (K) Tc (K)PdH 4.06 0.00 0.46 2.82PtH 2.73 0.00 0.44 2.82LaH10 3.84 0.00 0.00 0.00H2S 2.97 0.14 0.58 0.00H4Si 3.11 0.00 0.00 0.00Table 10 Ratio of proposed materials satisfying the following threeconditions: (1) hydrogen (H) content is 40% or more, (2) composed ofthree or fewer elements, and (3) 15 atoms or lessBase materialsfrom SuperCon KIAGO SD SD w/ CG C-ESPdH 1.00 0.03 0.04 1.00PtH 1.00 0.03 0.05 1.00LaH10 1.00 0.02 0.02 1.00H2S 1.00 0.04 0.05 1.00H4Si 1.00 0.03 0.02 1.00Digital Discovery PaperOpen Access Article. Published on 29 October 2025. Downloaded on 2/4/2026 7:46:00 AM.  This article is licensed under a Creative Commons Attribution 3.0 Unported Licence.View Article Onlinecharge neutrality, because SMACT assumes each element hasa single xed valence. Then, we assumed each atom of the sameelement could adopt different valences. For example, hydrogencan be both +1 and −1, making LaH10 = {La2+, H1− × 6, H1+ ×4}, which is thus electrically neutral. However, allowingmultiple valences can lead to a combinatorial explosion, so weimposed a maximum total of 15 atoms per composition. Weused these criteria (charge neutrality under variable valences,ternary or binary composition, and a total of 15 or fewer atoms)as a replacement for SMACT-based screening.3670 | Digital Discovery, 2025, 4, 3662–3673For initialisation for KIAGO, we began with HSC from theSuperCon dataset. With a probability of 0.29, we replacedelements of its composition with different elements chosenaccording to their occurrence frequencies in SuperCon. We thenselected random elements with random compositional ratios(from 0.0 to 0.03) for those elements, normalised the resultingcomposition, and used it as the initialisation. We also set {Nunit}= {1, 2, ., 15}.In Table 9, we present the average Tc of the top ve proposedhydride superconductors. Compared with other methods,KIAGO efficiently generates hydride superconductors. Table 10shows the probability of proposing materials that meet speciccriteria—namely, having at least 40% hydrogen content, threeor fewer elements, and a total atom count of 15 or below. Theseresults indicate that KIAGO not only explores the search spaceefficiently but also strictly adheres to the specied constraints.Table 11 lists HSC proposed by KIAGO. Notably, KIAGO alsoproposed materials made of the same elements as those inknown compounds from the SuperCon dataset. In addition, itsuggested materials that are not in the SuperCon dataset buthave been reported in other literature.4 LimitationOur method relies heavily on the accuracy of the predictionmodels. Although our current Tc predictor achieves competitiveperformance compared with other methods (see Section S.1.3),the predicted Tc values for the proposed materials inevitablycontain some error. We provide composition-level estimates (e.g.,ElemNet-based DEhull); however, DFT-validated, structure-dependent metrics remain unavailable. ElemNet was trained oncompositions with up to seven constituent elements, and itsability to generalise to systems with more elements is inherentlylimited, making such predictions partially extrapolative. Never-theless, as demonstrated in high-entropy alloy systems, moderate© 2025 The Author(s). Published by the Royal Society of Chemistryhttp://creativecommons.org/licenses/by/3.0/http://creativecommons.org/licenses/by/3.0/https://doi.org/10.1039/d5dd00250hTable 11 Candidates of hydride superconductor proposed by KIAGO. The ‘Similar formula in Refs’ and ‘Tc in Refs’ columns show the compositionand Tc of superconductors experimentally confirmed or calculated by DFT in other studies, respectively, which are composed of the sameelements as the proposed materialsProposed formulaPredictedTc (K)Similar formulain datasetTc (K)in SuperConSimilar formulain refs Tc (K) in refsSiH 0.7 SiH4 17.0 — —ZrH 0.5 — — ZrH3 6.7EXP61V3H2 1.8 — — VH 6.5 ∼ 10.7DFT62ScH 2.8 — — ScH2 38DFT63Paper Digital DiscoveryOpen Access Article. Published on 29 October 2025. Downloaded on 2/4/2026 7:46:00 AM.  This article is licensed under a Creative Commons Attribution 3.0 Unported Licence.View Article Onlineextrapolation can still yield reasonably accurate results.64 More-over, since the SuperCon dataset lacks pressure information—crucial for hydride superconductors—our model cannot addresspressure effects, which could pose another limitation.Improving the prediction model's accuracy must involveensemble methods,14 better model architectures, or enhanceddatasets. Importantly, ourmethod does not depend on any specicmodel architecture. Given the rapid pace of machine-learningadvances, more accurate models will likely become availablesoon, and substituting them into our framework should alleviatecurrent limitations. Additionally, datasets are also improving ata fast rate, offering further opportunities for renement. Incor-porating crystal structure prediction (CSP)65,66 from compositionmay mitigate the limitation of missing structural information,while multi-modal learning7,67 at the ne-tuning stage may enablemodels to consider essential factors such as pressure.5 ConclusionIn this paper, we introduced KIAGO, a gradient-basedmethod forproposing high-Tc superconductors that unify domain knowl-edge with efficient computational strategies. Unlike classierguidance-based generative models, KIAGO does not require totrain additional generative models, making it a more straight-forward solution. By initialising the optimisation from promisingsuperconductors, we mitigate the risk of converging to poor localminima—an issue oen encountered in gradient-basedmethods—and achieve higher optimisation efficiency. A keystrength of KIAGO lies in its ability to incorporate diverse domainknowledge via masking. We demonstrated this by preciselycontrolling elemental substitutions and restricting our search tohydride superconductors. These results underscore the adapt-ability of KIAGO: it not only capitalises on existing knowledge,like traditional doping strategies but also explores a broaderchemical space more effectively than previous approaches.Overall, KIAGO paves the way for discovering new materials byexploiting domain knowledge andmachine learning's scalability.This synergy has the potential to accelerate advancements inhigh-Tc superconductivity and beyond, offering a robust frame-work for rapid and adaptive materials design.Author contributionsA. F. performed the experiments and draed the manuscript;A. L. and K. S. contributed to discussions; S. W. supervised theresearch and provided project administration.© 2025 The Author(s). Published by the Royal Society of ChemistryConflicts of interestThere are no conicts to declare.Data availabilityThe code used to implement the optimisation framework in thisstudy is available at Zenodo: https://doi.org/10.5281/zenodo.17319338, which archives the GitHub repository athttps://github.com/AkiraTOSEI/KIAGO. This versioncorresponds to the release v0.3.0, accessed on 10 October2025. Trained models for the Tc predictor and formationenergy predictor are also provided in this repository. TheSuperCon dataset is available at https://mdr.nims.go.jp/collections/5712mb227 and we used version 220808. TheElemNet training data can be obtained from http://cucis.ece.northwestern.edu/projects/DataSets/ElemNet/data.tar.gz, and the COD database can be accessed at https://www.crystallography.net/cod/ and we accessed the CODdataset on March 22, 2023. The Alexandria Materials Databaseand the Materials Project were accessed on 8 October 2025.Supplementary information: details on predictive-modeltraining and performance, information on SuperDiff withClassier Guidance (SD w/ CG), and additional experimentalresults for KIAGO. See DOI: https://doi.org/10.1039/d5dd00250h.AcknowledgementsThe authors gratefully acknowledge support from the DoctoralStudent Special Incentives Program, Graduate School of Engi-neering, The University of Tokyo (SEUT-RA).Notes and references1 J. R. Hull, Rep. Prog. Phys., 2003, 66, 1865.2 D. Uglietti, Supercond. Sci. Technol., 2019, 32, 053001.3 J. Skakle, Mater. Sci. Eng., R, 1998, 23, 1–40.4 Y. Yao, C. Song, P. Bao, D. Su, X. Lu, J. Zhu and Y. Wang, J.Appl. Phys., 2004, 95, 3126–3130.5 S. C. Erwin, L. Zu, M. I. Hael, A. L. Efros, T. A. Kennedy andD. J. Norris, Nature, 2005, 436, 91–94.6 R. Terzioglu, G. Aydin, N. Soylu Koc and C. Terzioglu, J.Mater. Sci.: Mater. Electron., 2019, 30, 2265–2277.7 V. Stanev, C. Oses, A. G. Kusne, E. Rodriguez, J. Paglione,S. Curtarolo and I. Takeuchi, npj Comput. Mater., 2018, 4, 29.Digital Discovery, 2025, 4, 3662–3673 | 3671https://doi.org/10.5281/zenodo.17319338https://doi.org/10.5281/zenodo.17319338https://github.com/AkiraTOSEI/KIAGOhttps://mdr.nims.go.jp/collections/5712mb227https://mdr.nims.go.jp/collections/5712mb227http://cucis.ece.northwestern.edu/projects/DataSets/ElemNet/data.tar.gzhttp://cucis.ece.northwestern.edu/projects/DataSets/ElemNet/data.tar.gzhttp://cucis.ece.northwestern.edu/projects/DataSets/ElemNet/data.tar.gzhttps://www.crystallography.net/cod/https://www.crystallography.net/cod/https://doi.org/10.1039/d5dd00250hhttps://doi.org/10.1039/d5dd00250hhttp://creativecommons.org/licenses/by/3.0/http://creativecommons.org/licenses/by/3.0/https://doi.org/10.1039/d5dd00250hDigital Discovery PaperOpen Access Article. Published on 29 October 2025. Downloaded on 2/4/2026 7:46:00 AM.  This article is licensed under a Creative Commons Attribution 3.0 Unported Licence.View Article Online8 K. Matsumoto and T. Horide, Appl. Phys. Express, 2019, 12,073003.9 S. Zeng, Y. Zhao, G. Li, R. Wang, X. Wang and J. Ni, npjComput. Mater., 2019, 5, 84.10 Y. Dan, R. Dong, Z. Cao, X. Li, C. Niu, S. Li and J. Hu, IEEEAccess, 2020, 8, 57868–57878.11 T. D. Le, R. Noumeir, H. L. Quach, J. H. Kim, J. H. Kim andH. M. Kim, IEEE Trans. Appl. Supercond., 2020, 30, 1–5.12 T. Konno, H. Kurokawa, F. Nabeshima, Y. Sakishita,R. Ogawa, I. Hosako and A. Maeda, Phys. Rev. B, 2021, 103,014509.13 J. Zhang, Z. Zhu, X.-D. Xiang, K. Zhang, S. Huang, C. Zhong,H.-J. Qiu, K. Hu and X. Lin, J. Phys. Chem. C, 2022, 126, 8922–8927.14 A. Taheri, H. Ebrahimnezhad and M. H. Sedaaghi, Mater.Today Commun., 2022, 33, 104743.15 J. Zhang, K. Zhang, S. Xu, Y. Li, C. Zhong, M. Zhao, H.-J. Qiu,M. Qin, X.-D. Xiang, K. Hu, et al., J. Energy Chem., 2023, 78,232–239.16 N. I. for Materials Science, SuperCon, version 220808, https://mdr.nims.go.jp/collections/5712mb227.17 C. Pereti, K. Bernot, T. Guizouarn, F. Laufek, A. Vymazalová,L. Bindi, R. Sessoli and D. Fanelli, npj Comput. Mater., 2023,9, 71.18 E. A. Pogue, A. New, K. McElroy, N. Q. Le, M. J. Pekala,I. McCue, E. Gienger, J. Domenico, E. Hedrick,T. M. McQueen, et al., npj Comput. Mater., 2023, 9, 181.19 D. Kaplan, A. Zheng, J. Blawat, R. Jin, R. J. Cava,V. Oudovenko, G. Kotliar, A. M. Sengupta and W. Xie, Eur.Phys. J. Plus, 2025, 140, 58.20 T. Xie, X. Fu, O.-E. Ganea, R. Barzilay and T. Jaakkola, arXiv,2021, preprint, arXiv:2110.06197, DOI: 10.48550/arXiv.2110.06197.21 Z. Ren, S. I. P. Tian, J. Noh, F. Oviedo, G. Xing, J. Li, Q. Liang,R. Zhu, A. G. Aberle, S. Sun, et al., Matter, 2022, 5, 314–335.22 C. Zeni, R. Pinsler, D. Zügner, A. Fowler, M. Horton, X. Fu,S. Shysheya, J. Crabbé, L. Sun and J. Smith, et al., arXiv,2023, preprint, arXiv:2312.03687, DOI: 10.48550/arXiv.2312.03687.23 Z. Yang, W. Ye, X. Lei, D. Schweigert, H.-K. Kwon andA. Khajeh, npj Comput. Mater., 2024, 10, 296.24 C. Zhong, J. Zhang, X. Lu, K. Zhang, J. Liu, K. Hu, J. Chen andX. Lin, ACS Appl. Mater. Interfaces, 2023, 15, 30029–30038.25 E. Kim and S. Dordevic, J. Phys.: Condens. Matter, 2023, 36,025702.26 D. Wines, T. Xie and K. Choudhary, J. Phys. Chem. Lett., 2023,14, 6630–6638.27 S. Yuan and S. Dordevic, Sci. Rep., 2024, 14, 10275.28 J. Ho, A. Jain and P. Abbeel, Adv. Neural Inf. Process. Syst.,2020, 33, 6840–6851.29 F.-A. Croitoru, V. Hondru, R. T. Ionescu and M. Shah, IEEETrans. Pattern Anal. Mach. Intell., 2023, 45, 10850–10869.30 J. Choi, S. Kim, Y. Jeong, Y. Gwon and S. Yoon, arXiv, 2021,preprint, arXiv:2108.02938, DOI: 10.48550/arXiv.2108.02938.31 A. Brock, J. Donahue and K. Simonyan, arXiv, 2018, preprint,arXiv:1809.11096, DOI: 10.48550/arXiv.1809.11096.3672 | Digital Discovery, 2025, 4, 3662–367332 A. Bansal, H.-M. Chu, A. Schwarzschild, S. Sengupta,M. Goldblum, J. Geiping and T. Goldstein, Proceedings ofthe IEEE/CVF Conference on Computer Vision and PatternRecognition (CVPR) Workshops, 2023, pp. 843–852.33 P. Dhariwal and A. Nichol, Adv. Neural Inf. Process. Syst.,2021, 34, 8780–8794.34 J. Ho and T. Salimans, arXiv, 2022, preprint,arXiv:2207.12598, DOI: 10.48550/arXiv.2207.12598.35 S. Ren, W. Padilla and J. Malof, Adv. Neural Inf. Process. Syst.,2020, 33, 38–48.36 K. R. Allen, T. Lopez-Guevara, K. Stachenfeld, A. Sanchez-Gonzalez, P. Battaglia, J. Hamrick and T. Pfaff, arXiv, 2022,preprint, arXiv:2202.00728, DOI: 10.48550/arXiv.2202.00728.37 R. Hwang, J. Y. Lee, J. Y. Shin and H. J. Hwang, Proceedings ofthe AAAI Conference on Articial Intelligence, 2022, pp. 4504–4512.38 A. Fujii, H. Tsunashima, Y. Fukuhara, K. Shimizu andS. Watanabe, arXiv, 2023, preprint, arXiv:2304.13860, DOI:10.48550/arXiv.2304.13860.39 A. Paszke, S. Gross, S. Chintala, G. Chanan, E. Yang,Z. DeVito, Z. Lin, A. Desmaison, L. Antiga and A. Lerer,NIPS 2017 Workshop Autodiff, 2017.40 D. P. Kingma and J. Ba, arXiv, 2014, preprint,arXiv:1412.6980, DOI: 10.48550/arXiv.1412.6980.41 K. He, X. Zhang, S. Ren and J. Sun, Proceedings of the IEEEconference on computer vision and pattern recognition, 2016,pp. 770–778.42 Crystallography Open Database (COD), https://www.crystallography.net/cod/, accessed: March 22, 2023.43 A. Vaitkus, A. Merkys, T. Sander, M. Quirós, P. A. Thiessen,E. E. Bolton and S. Gražulis, J. Cheminf., 2023, 15, 123.44 A. Merkys, A. Vaitkus, A. Grybauskas, A. Konovalovas,M. Quirós and S. Gražulis, J. Cheminf., 2023, 15, 25.45 A. Vaitkus, A. Merkys and S. Gražulis, Appl. Crystallogr., 2021,54, 661–672.46 M. Quirós, S. Gražulis, S. Girdzijauskaitė, A. Merkys andA. Vaitkus, J. Cheminf., 2018, 10, 23.47 A. Merkys, A. Vaitkus, J. Butkus, M. Okulič-Kazarinas,V. Kairys and S. Gražulis, Appl. Crystallogr., 2016, 49, 292–301.48 S. Gražulis, A. Merkys, A. Vaitkus and M. Okulič-Kazarinas,Appl. Crystallogr., 2015, 48, 85–91.49 S. Gražulis, A. Daškevič, A. Merkys, D. Chateigner,L. Lutterotti, M. Quiros, N. R. Serebryanaya, P. Moeck,R. T. Downs and A. Le Bail, Nucleic Acids Res., 2012, 40,D420–D427.50 S. Gražulis, D. Chateigner, R. T. Downs, A. Yokochi,M. Quirós, L. Lutterotti, E. Manakova, J. Butkus, P. Moeckand A. Le Bail, Appl. Crystallogr., 2009, 42, 726–729.51 R. T. Downs and M. Hall-Wallace, Am. Mineral., 2003, 88,247–250.52 D. Jha, L. Ward, A. Paul, W.-k. Liao, A. Choudhary,C. Wolverton and A. Agrawal, Sci. Rep., 2018, 8, 17593.53 M. Abadi, A. Agarwal, P. Barham, E. Brevdo, Z. Chen,C. Citro, G. S. Corrado, A. Davis, J. Dean, M. Devin,S. Ghemawat, I. Goodfellow, A. Harp, G. Irving, M. Isard,Y. Jia, R. Jozefowicz, L. Kaiser, M. Kudlur, J. Levenberg,© 2025 The Author(s). Published by the Royal Society of Chemistryhttps://mdr.nims.go.jp/collections/5712mb227https://mdr.nims.go.jp/collections/5712mb227https://doi.org/10.48550/arXiv.2110.06197https://doi.org/10.48550/arXiv.2110.06197https://doi.org/10.48550/arXiv.2312.03687https://doi.org/10.48550/arXiv.2312.03687https://doi.org/10.48550/arXiv.2108.02938https://doi.org/10.48550/arXiv.1809.11096https://doi.org/10.48550/arXiv.2207.12598https://doi.org/10.48550/arXiv.2202.00728https://doi.org/10.48550/arXiv.2304.13860https://doi.org/10.48550/arXiv.1412.6980https://www.crystallography.net/cod/https://www.crystallography.net/cod/http://creativecommons.org/licenses/by/3.0/http://creativecommons.org/licenses/by/3.0/https://doi.org/10.1039/d5dd00250hPaper Digital DiscoveryOpen Access Article. Published on 29 October 2025. Downloaded on 2/4/2026 7:46:00 AM.  This article is licensed under a Creative Commons Attribution 3.0 Unported Licence.View Article OnlineD. Mané, R. Monga, S. Moore, D. Murray, C. Olah,M. Schuster, J. Shlens, B. Steiner, I. Sutskever, K. Talwar,P. Tucker, V. Vanhoucke, V. Vasudevan, F. Viégas,O. Vinyals, P. Warden, M. Wattenberg, M. Wicke, Y. Yuand X. Zheng, TensorFlow: Large-Scale Machine Learning onHeterogeneous Systems, 2015, https://www.tensorow.org/,Soware available from tensorow.org.54 D. W. Davies, K. T. Butler, A. J. Jackson, J. M. Skelton,K. Morita and A. Walsh, J. Open Source Sow., 2019, 4, 1361.55 M. K. Horton, P. Huck, R. X. Yang, J. M. Munro,S. Dwaraknath, A. M. Ganose, R. S. Kingsbury, M. Wen,J. X. Shen, T. S. Mathis, et al., Nat. Mater., 2025, 1–11.56 Alexandria Materials Database, https://alexandria.icams.rub.de/, accessed: September 28, 2025.57 J. Schmidt, L. Pettersson, C. Verdozzi, S. Botti andM. A. Marques, Sci. Adv., 2021, 7, eabi7948.58 J. Schmidt, N. Hoffmann, H.-C. Wang, P. Borlido,P. J. Carriço, T. F. Cerqueira, S. Botti and M. A. Marques,Adv. Mater., 2023, 35, 2210788.59 H.-C. Wang, J. Schmidt, M. A. Marques, L. Wirtz andA. H. Romero, 2D Materials, 2023, 10, 035007.© 2025 The Author(s). Published by the Royal Society of Chemistry60 S. P. Ong, W. D. Richards, A. Jain, G. Hautier, M. Kocher,S. Cholia, D. Gunter, V. L. Chevrier, K. A. Persson andG. Ceder, Comput. Mater. Sci., 2013, 68, 314–319.61 H. Xie, W. Zhang, D. Duan, X. Huang, Y. Huang, H. Song,X. Feng, Y. Yao, C. J. Pickard and T. Cui, J. Phys. Chem.Lett., 2020, 11, 646–651.62 Q. Zhuang, X. Jin, Q. Lv, Y. Li, Z. Shao, Z. Liu, X. Li, H. Zhang,X. Meng, K. Bao and T. Cui, Phys. Chem. Chem. Phys., 2017,19, 26280–26284.63 Y.-K. Wei, J.-N. Yuan, F. I. Khan, G.-F. Ji, Z.-W. Gu andD.-Q. Wei, RSC Adv., 2016, 6, 81534–81541.64 K. Miyamoto, K. Shimizu, A. K. A. Lu and S. Watanabe, e-J.Surf. Sci. Nanotechnol., 2025, 23, 188–192.65 S. Kim, J. Noh, G. H. Gu, A. Aspuru-Guzik and Y. Jung, ACSCent. Sci., 2020, 6, 1412–1420.66 R. Jiao, W. Huang, P. Lin, J. Han, P. Chen, Y. Lu and Y. Liu,Adv. Neural Inf. Process. Syst., 2023, 36, 17464–17497.67 C. Chen, W. Ye, Y. Zuo, C. Zheng and S. P. Ong, Chem. Mater.,2019, 31, 3564–3572.Digital Discovery, 2025, 4, 3662–3673 | 3673https://www.tensorflow.org/https://alexandria.icams.rub.de/https://alexandria.icams.rub.de/http://creativecommons.org/licenses/by/3.0/http://creativecommons.org/licenses/by/3.0/https://doi.org/10.1039/d5dd00250h A straightforward gradient-based approach for designing superconductors with high critical temperature: exploiting domain knowledge via adaptive constraints A straightforward gradient-based approach for designing superconductors with high critical temperature: exploiting domain knowledge via adaptive constraints A straightforward gradient-based approach for designing superconductors with high critical temperature: exploiting domain knowledge via adaptive constraints A straightforward gradient-based approach for designing superconductors with high critical temperature: exploiting domain knowledge via adaptive constraints A straightforward gradient-based approach for designing superconductors with high critical temperature: exploiting domain knowledge via adaptive constraints A straightforward gradient-based approach for designing superconductors with high critical temperature: exploiting domain knowledge via adaptive constraints A straightforward gradient-based approach for designing superconductors with high critical temperature: exploiting domain knowledge via adaptive constraints A straightforward gradient-based approach for designing superconductors with high critical temperature: exploiting domain knowledge via adaptive constraints A straightforward gradient-based approach for designing superconductors with high critical temperature: exploiting domain knowledge via adaptive constraints A straightforward gradient-based approach for designing superconductors with high critical temperature: exploiting domain knowledge via adaptive constraints A straightforward gradient-based approach for designing superconductors with high critical temperature: exploiting domain knowledge via adaptive constraints A straightforward gradient-based approach for designing superconductors with high critical temperature: exploiting domain knowledge via adaptive constraints A straightforward gradient-based approach for designing superconductors with high critical temperature: exploiting domain knowledge via adaptive constraints A straightforward gradient-based approach for designing superconductors with high critical temperature: exploiting domain knowledge via adaptive constraints A straightforward gradient-based approach for designing superconductors with high critical temperature: exploiting domain knowledge via adaptive constraints A straightforward gradient-based approach for designing superconductors with high critical temperature: exploiting domain knowledge via adaptive constraints A straightforward gradient-based approach for designing superconductors with high critical temperature: exploiting domain knowledge via adaptive constraints A straightforward gradient-based approach for designing superconductors with high critical temperature: exploiting domain knowledge via adaptive constraints A straightforward gradient-based approach for designing superconductors with high critical temperature: exploiting domain knowledge via adaptive constraints A straightforward gradient-based approach for designing superconductors with high critical temperature: exploiting domain knowledge via adaptive constraints A straightforward gradient-based approach for designing superconductors with high critical temperature: exploiting domain knowledge via adaptive constraints A straightforward gradient-based approach for designing superconductors with high critical temperature: exploiting domain knowledge via adaptive constraints A straightforward gradient-based approach for designing superconductors with high critical temperature: exploiting domain knowledge via adaptive constraints