{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,9,28]],"date-time":"2026-09-28T02:22:43Z","timestamp":1790562163427,"version":"4.1.0"},"reference-count":31,"publisher":"MDPI AG","issue":"5","license":[{"start":{"date-parts":[[2023,5,9]],"date-time":"2023-05-09T00:00:00Z","timestamp":1683590400000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"Deutsche Forschungsgemeinschaft","award":["390685689"],"award-info":[{"award-number":["390685689"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Computation"],"abstract":"<jats:p>Predicting the chemical stability of yet-to-be-discovered materials is an important aspect of the discovery and development of virtual materials. The conventional approach for computing the enthalpy of formation based on ab initio methods is time consuming and computationally demanding. In this regard, alternative machine learning approaches are proposed to predict the formation energies of different classes of materials with decent accuracy. In this paper, one such machine learning approach, a novel two-step method that predicts the formation energy of ternary compounds, is presented. In the first step, with a classifier, we determine the accuracy of heuristically calculated formation energies in order to increase the size of the training dataset for the second step. The second step is a regression model that predicts the formation energy of the ternary compounds. The first step leads to at least a 100% increase in the size of the dataset with respect to the data available in the Materials Project database. The results from the regression model match those from the existing state-of-the-art prediction models. In addition, we propose a slightly modified version of the Adam optimizer, namely centered Adam, and report the results from testing the centered Adam optimizer.<\/jats:p>","DOI":"10.3390\/computation11050095","type":"journal-article","created":{"date-parts":[[2023,5,10]],"date-time":"2023-05-10T03:49:06Z","timestamp":1683690546000},"page":"95","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":4,"title":["A Two-Step Machine Learning Method for Predicting the Formation Energy of Ternary Compounds"],"prefix":"10.3390","volume":"11","author":[{"given":"Varadarajan","family":"Rengaraj","sequence":"first","affiliation":[{"name":"Dynamics of Condensed Matter, Chair of Theoretical Chemistry, University of Paderborn, Warburger Str. 100, 33098 Paderborn, Germany"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0009-0005-3185-5903","authenticated-orcid":false,"given":"Sebastian","family":"Jost","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Humboldt-Universit\u00e4t zu Berlin, Unter den Linden 6, 10099 Berlin, Germany"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-0553-4242","authenticated-orcid":false,"given":"Franz","family":"Bethke","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Humboldt-Universit\u00e4t zu Berlin, Unter den Linden 6, 10099 Berlin, Germany"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-5728-9982","authenticated-orcid":false,"given":"Christian","family":"Plessl","sequence":"additional","affiliation":[{"name":"Department of Computer Science, University of Paderborn, Warburger Str. 100, 33098 Paderborn, Germany"},{"name":"Paderborn Center for Parallel Computing (PC2), University of Paderborn, Warburger Str. 100, 33098 Paderborn, Germany"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-6179-1545","authenticated-orcid":false,"given":"Hossein","family":"Mirhosseini","sequence":"additional","affiliation":[{"name":"Dynamics of Condensed Matter, Chair of Theoretical Chemistry, University of Paderborn, Warburger Str. 100, 33098 Paderborn, Germany"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-3516-4641","authenticated-orcid":false,"given":"Andrea","family":"Walther","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Humboldt-Universit\u00e4t zu Berlin, Unter den Linden 6, 10099 Berlin, Germany"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-5471-2407","authenticated-orcid":false,"given":"Thomas D.","family":"K\u00fchne","sequence":"additional","affiliation":[{"name":"Dynamics of Condensed Matter, Chair of Theoretical Chemistry, University of Paderborn, Warburger Str. 100, 33098 Paderborn, Germany"},{"name":"Paderborn Center for Parallel Computing (PC2), University of Paderborn, Warburger Str. 100, 33098 Paderborn, Germany"},{"name":"Center for Sustainable Systems Design, University of Paderborn, Warburger Str. 100, 33098 Paderborn, Germany"}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"1968","published-online":{"date-parts":[[2023,5,9]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"296","DOI":"10.1088\/0957-0233\/16\/1\/039","article-title":"High-throughput and data mining with ab initio methods","volume":"16","author":"Morgan","year":"2004","journal-title":"Meas. Sci. Technol."},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"454","DOI":"10.1016\/j.commatsci.2018.04.033","article-title":"Predicting the thermodynamic stability of perovskite oxides using machine learning models","volume":"150","author":"Li","year":"2018","journal-title":"Comput. Mater. Sci."},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"241728","DOI":"10.1063\/1.5020223","article-title":"Predicting the stability of ternary intermetallics with density functional theory and machine learning","volume":"148","author":"Schmidt","year":"2018","journal-title":"J. Chem. Phys."},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"617","DOI":"10.1016\/j.chempr.2016.09.010","article-title":"Computational Screening of All Stoichiometric Inorganic Materials","volume":"1","author":"Davies","year":"2016","journal-title":"Chem"},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"094104","DOI":"10.1103\/PhysRevB.89.094104","article-title":"Combinatorial screening for new materials in unconstrained composition space with machine learning","volume":"89","author":"Meredig","year":"2014","journal-title":"Phys. Rev. B"},{"key":"ref_6","unstructured":"Liu, R., Ward, L., Wolverton, C., Agrawal, A., Liao, W., and Choudhary, A. (2016, January 13\u201317). Deep learning for chemical compound stability prediction. Proceedings of the ACM SIGKDD Workshop on Large-Scale Deep Learning for Data Mining (DL-KDD), San Francisco, CA, USA."},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"17593","DOI":"10.1038\/s41598-018-35934-y","article-title":"ElemNet: Deep Learning the Chemistry of Materials From Only Elemental Composition","volume":"8","author":"Jha","year":"2018","journal-title":"Sci. Rep."},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"145301","DOI":"10.1103\/PhysRevLett.120.145301","article-title":"Crystal Graph Convolutional Neural Networks for an Accurate and Interpretable Prediction of Material Properties","volume":"120","author":"Xie","year":"2018","journal-title":"Phys. Rev. Lett."},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"448","DOI":"10.1021\/acs.jctc.8b00908","article-title":"SchNetPack: A Deep Learning Toolbox For Atomistic Systems","volume":"15","author":"Kessel","year":"2019","journal-title":"J. Chem. Theory Comput."},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"185","DOI":"10.1038\/s41524-021-00650-1","article-title":"Atomistic Line Graph Neural Network for improved materials property predictions","volume":"7","author":"Choudhary","year":"2021","journal-title":"npj Comput. Mater."},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"4954","DOI":"10.1021\/acs.chemmater.0c01907","article-title":"Machine Learning for Materials Scientists: An Introductory Guide toward Best Practices","volume":"32","author":"Wang","year":"2020","journal-title":"Chem. Mater."},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"022002","DOI":"10.1088\/2515-7655\/abe425","article-title":"Materials discovery through machine learning formation energy","volume":"3","author":"Peterson","year":"2021","journal-title":"J. Phys. Energy"},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"15010","DOI":"10.1038\/npjcompumats.2015.10","article-title":"The Open Quantum Materials Database (OQMD): Assessing the accuracy of DFT formation energies","volume":"1","author":"Kirklin","year":"2015","journal-title":"npj Comput. Mater."},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"93","DOI":"10.1038\/s43246-022-00315-6","article-title":"Graph neural networks for materials science and chemistry","volume":"3","author":"Reiser","year":"2022","journal-title":"Commun. Mater."},{"key":"ref_15","unstructured":"Yan, K., Liu, Y., Lin, Y., and Ji, S. (December, January 28). Periodic Graph Transformers for Crystal Material Property Prediction. Proceedings of the 36th Annual Conference on Neural Information Processing Systems, New Orleans, LA, USA."},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"011002","DOI":"10.1063\/1.4812323","article-title":"The Materials Project: A materials genome approach to accelerating materials innovation","volume":"1","author":"Jain","year":"2013","journal-title":"APL Mater."},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"3564","DOI":"10.1021\/acs.chemmater.9b01294","article-title":"Graph Networks as a Universal Machine Learning Framework for Molecules and Crystals","volume":"31","author":"Chen","year":"2019","journal-title":"Chem. Mater."},{"key":"ref_18","doi-asserted-by":"crossref","unstructured":"Hillert, M. (2007). Phase Equilibria, Phase Diagrams and Phase Transformations: Their Thermodynamic Basis, Cambridge University Press. [2nd ed.].","DOI":"10.1017\/CBO9780511812781"},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"83","DOI":"10.1051\/jcp\/1975720083","article-title":"Enthalpies of formation of liquid alloys bismuth-gallium-tin at 723k-choice of an analytical representation of integral and partial thermodynamic functions of mixing for this ternary-system","volume":"72","author":"Muggianu","year":"1975","journal-title":"J. Chim. Phys. Phys.-Chim. Biol."},{"key":"ref_20","unstructured":"Kingma, D.P., and Ba, J. (2015, January 7\u20139). Adam: A Method for Stochastic Optimization. Proceedings of the 3rd International Conference on Learning Representations, ICLR 2015, San Diego, CA, USA. Conference Track Proceedings; 2015."},{"key":"ref_21","unstructured":"Jost, S. (2023, May 04). Comparison of Architectures and Parameters for Artificial Neural Networks. Available online: https:\/\/github.com\/simulatedScience\/Bachelorthesis."},{"key":"ref_22","unstructured":"Chollet, F. (2023, May 04). Keras. Available online: https:\/\/keras.io."},{"key":"ref_23","unstructured":"Abadi, M., Agarwal, A., Barham, P., Brevdo, E., Chen, Z., Citro, C., Corrado, G.S., Davis, A., Dean, J., and Devin, M. (2023, May 04). Available online: https:\/\/www.tensorflow.org\/."},{"key":"ref_24","first-page":"2825","article-title":"Scikit-learn: Machine Learning in Python","volume":"12","author":"Pedregosa","year":"2011","journal-title":"J. Mach. Learn. Res."},{"key":"ref_25","doi-asserted-by":"crossref","first-page":"22","DOI":"10.1109\/MCSE.2011.37","article-title":"The NumPy array: A structure for efficient numerical computation","volume":"13","author":"Colbert","year":"2011","journal-title":"Comput. Sci. Eng."},{"key":"ref_26","unstructured":"Jones, E., Oliphant, T., and Peterson, P. (2023, May 04). SciPy: Open Source Scientific Tools for Python. Available online: http:\/\/www.scipy.org."},{"key":"ref_27","unstructured":"McKinney, W. (July, January 28). Data Structures for Statistical Computing in Python. Proceedings of the 9th Python in Science Conference, Austin, TX, USA."},{"key":"ref_28","doi-asserted-by":"crossref","first-page":"90","DOI":"10.1109\/MCSE.2007.55","article-title":"Matplotlib: A 2D graphics environment","volume":"9","author":"Hunter","year":"2007","journal-title":"Comput. Sci. Eng."},{"key":"ref_29","unstructured":"Rengaraj, V. (2019). A Two-Step Machine Learning for Predicting the Stability of Chemical Compositions. [Master\u2019s Thesis, Paderborn University]."},{"key":"ref_30","unstructured":"LeCun, Y., and Cortes, C. (2023, May 04). MNIST Handwritten Digit Database. Available online: http:\/\/yann.lecun.com\/exdb\/mnist\/."},{"key":"ref_31","unstructured":"Li, C., Farkhoor, H., Liu, R., and Yosinski, J. (May, January 30). Measuring the Intrinsic Dimension of Objective Landscapes. Proceedings of the 6th International Conference on Learning Representations, ICLR 2018, Vancouver, BC, Canada."}],"container-title":["Computation"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2079-3197\/11\/5\/95\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,10]],"date-time":"2025-10-10T19:32:29Z","timestamp":1760124749000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2079-3197\/11\/5\/95"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2023,5,9]]},"references-count":31,"journal-issue":{"issue":"5","published-online":{"date-parts":[[2023,5]]}},"alternative-id":["computation11050095"],"URL":"https:\/\/doi.org\/10.3390\/computation11050095","relation":{},"ISSN":["2079-3197"],"issn-type":[{"value":"2079-3197","type":"electronic"}],"subject":[],"published":{"date-parts":[[2023,5,9]]}}}