{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,12,5]],"date-time":"2025-12-05T19:28:14Z","timestamp":1764962894498,"version":"3.46.0"},"reference-count":39,"publisher":"MDPI AG","issue":"12","license":[{"start":{"date-parts":[[2025,11,28]],"date-time":"2025-11-28T00:00:00Z","timestamp":1764288000000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"Ministerio de Ciencia, Innovaci\u00f3n y Universidades, Gobierno de Espa\u00f1a","award":["PID2024-57499OB-C31"],"award-info":[{"award-number":["PID2024-57499OB-C31"]}]},{"name":"Consejer\u00eda de Universidades, Ciencia e Innovaci\u00f3n y Cultura del Gobierno de Canarias","award":["PRECOMP02 SD-24\/03"],"award-info":[{"award-number":["PRECOMP02 SD-24\/03"]}]},{"name":"Ministry of Universities of Spanish Government","award":["FPU22\/01933"],"award-info":[{"award-number":["FPU22\/01933"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Algorithms"],"abstract":"<jats:p>Surrogate models are widely used in science and engineering to approximate other methods that are usually computationally expensive. Here, artificial neural networks (ANNs) are employed as surrogate regression models to approximate the finite element method in the problem of structural analysis of steel frames. The focus is on a multi-objective neural architecture search (NAS) that minimizes the training time and maximizes the surrogate accuracy. To this end, several configurations of the non-dominated sorting genetic algorithm (NSGA-II) are tested versus random search. The robustness of the methodology is demonstrated by the statistical significance of the hypervolume indicator. Non-dominated solutions (consisting of the set of best designs in terms of accuracy for each training time or in terms of training time for each accuracy) reveal the importance of multi-objective hyperparameter tuning in the performance of ANNs as regression surrogates. Non-evident optimal values were attained for the number of hidden layers, the number of nodes per layer, the batch size, and alpha parameter of the Leaky ReLU transfer function. These results are useful for comparing with state-of-the-art ANN regression surrogates recently attained in the recent structural engineering literature. This approach facilitates the selection of models that achieve the optimal balance between training speed and predictive accuracy, according to the specific requirements of the application.<\/jats:p>","DOI":"10.3390\/a18120754","type":"journal-article","created":{"date-parts":[[2025,12,5]],"date-time":"2025-12-05T18:42:02Z","timestamp":1764960122000},"page":"754","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["A Multi-Objective Evolutionary Computation Approach for Improving Neural Network-Based Surrogate Models in Structural Engineering"],"prefix":"10.3390","volume":"18","author":[{"given":"N\u00e9stor","family":"L\u00f3pez-Gonz\u00e1lez","sequence":"first","affiliation":[{"name":"Institute of Intelligent Systems and Numerical Applications in Engineering (SIANI), University of Las Palmas de Gran Canaria (ULPGC), 35017 Las Palmas de Gran Canaria, Spain"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-2701-2971","authenticated-orcid":false,"given":"Eduardo","family":"Rodr\u00edguez","sequence":"additional","affiliation":[{"name":"Institute of Intelligent Systems and Numerical Applications in Engineering (SIANI), University of Las Palmas de Gran Canaria (ULPGC), 35017 Las Palmas de Gran Canaria, Spain"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-4132-7144","authenticated-orcid":false,"given":"David","family":"Greiner","sequence":"additional","affiliation":[{"name":"Institute of Intelligent Systems and Numerical Applications in Engineering (SIANI), University of Las Palmas de Gran Canaria (ULPGC), 35017 Las Palmas de Gran Canaria, Spain"}]}],"member":"1968","published-online":{"date-parts":[[2025,11,28]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"703","DOI":"10.1007\/s11831-016-9187-y","article-title":"Game theory based Evolutionary Algorithms: A review with Nash applications in structural engineering optimization problems","volume":"24","author":"Greiner","year":"2017","journal-title":"Arch. Comput. Methods Eng."},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"657","DOI":"10.1016\/0045-7949(91)90178-O","article-title":"Neurobiological computational models in structural analysis and design","volume":"41","author":"Hajela","year":"1991","journal-title":"Comput. Struct."},{"doi-asserted-by":"crossref","unstructured":"Topping, B.H.V. (1996). Some recent and current problems of neurocomputing in civil and structural engineering. Advances in Computational Structures Technology, CIVIL-COMP Press.","key":"ref_3","DOI":"10.4203\/ccp.38"},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"309","DOI":"10.1016\/S0045-7825(97)00215-6","article-title":"Structural optimization using evolution strategies and neural networks","volume":"156","author":"Papadrakakis","year":"1998","journal-title":"Comput. Methods Appl. Mech. Eng."},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"436","DOI":"10.1038\/nature14539","article-title":"Deep learning","volume":"521","author":"LeCun","year":"2015","journal-title":"Nature"},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"81","DOI":"10.1162\/evco_a_00325","article-title":"Evolutionary algorithms for parameter optimization\u2014Thirty years later","volume":"31","author":"Kononova","year":"2023","journal-title":"Evol. Comput."},{"unstructured":"Deb, K. (2001). Multi-Objective Optimization Using Evolutionary Algorithms, John Wiley & Sons.","key":"ref_7"},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"127472","DOI":"10.1016\/j.eswa.2025.127472","article-title":"A computational model for multiobjective optimization of multipolar stimulation in cochlear implants: An enhanced focusing approach","volume":"280","author":"Greiner","year":"2025","journal-title":"Expert Syst. Appl."},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"100478","DOI":"10.1016\/j.dibe.2024.100478","article-title":"Enhancing the maintenance strategy and cost in systems with surrogate assisted multiobjective evolutionary algorithms","volume":"19","author":"Greiner","year":"2024","journal-title":"Dev. Built Environ."},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"119495","DOI":"10.1016\/j.eswa.2022.119495","article-title":"A review of surrogate-assisted evolutionary algorithms for expensive optimization problems","volume":"217","author":"He","year":"2023","journal-title":"Expert Syst. Appl."},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"117928","DOI":"10.1016\/j.cma.2025.117928","article-title":"Deep learning-based surrogate capacity models and multi-objective fragility estimates for reinforced concrete frames","volume":"440","author":"Xing","year":"2025","journal-title":"Comput. Methods Appl. Mech. Eng."},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"106917","DOI":"10.1016\/j.compstruc.2022.106917","article-title":"Surrogate model based on ANN for the evaluation of the fundamental frequency of offshore wind turbines supported on jackets","volume":"274","year":"2023","journal-title":"Comput. Struct."},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"119984","DOI":"10.1016\/j.oceaneng.2024.119984","article-title":"ANN-based surrogate model for the structural evaluation of jacket support structures for offshore wind turbines","volume":"317","year":"2025","journal-title":"Ocean Eng."},{"key":"ref_14","first-page":"1","article-title":"Neural Architecture Search: A Survey","volume":"20","author":"Elsken","year":"2019","journal-title":"J. Mach. Learn. Res."},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"73","DOI":"10.1007\/s10462-024-11058-w","article-title":"Systematic review of neural architecture search","volume":"58","author":"Eskue","year":"2025","journal-title":"Artif. Intell. Rev."},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"609","DOI":"10.1016\/j.istruc.2023.04.006","article-title":"Metamodel-assisted design optimization in the field of structural engineering: A literature review","volume":"52","author":"Negrin","year":"2023","journal-title":"Structures"},{"doi-asserted-by":"crossref","unstructured":"Etim, B., Al-Ghosoun, A., Renno, J., Seaid, M., and Mohamed, M.S. (2024). Machine Learning-Based Modeling for Structural Engineering: A Comprehensive Survey and Applications Overview. Buildings, 14.","key":"ref_17","DOI":"10.3390\/buildings14113515"},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"142","DOI":"10.1007\/s42417-024-01749-7","article-title":"A Review on Artificial Neural Networks for Structural Analysis","volume":"13","author":"Saini","year":"2025","journal-title":"J. Vib. Eng. Technol."},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"735","DOI":"10.1007\/s11831-024-10152-0","article-title":"Application of Data-Driven Surrogate Models in Structural Engineering: A Literature Review","volume":"32","author":"Samadian","year":"2025","journal-title":"Arch. Comput. Methods Eng."},{"unstructured":"Kim, Y.H., Reddy, B., Yun, S., and Seo, C. (2017, January 6\u201311). NEMO: Neuro-evolution with multi-objective optimization of deep neural network for speed and accuracy. Proceedings of the International Conference on Machine Learning ICML 2017, Auto Machine Learning Workshop, Sydney, Australia.","key":"ref_20"},{"unstructured":"White, C., Safari, M., Sukthanker, R., Ru, B., Elsken, T., Zela, A., Dey, D., and Hutter, F. (2021). Neural Architecture Search: Insights from 1000 Papers. arXiv.","key":"ref_21"},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"182","DOI":"10.1109\/4235.996017","article-title":"A fast and elitist multiobjective genetic algorithm: NSGA-II","volume":"6","author":"Deb","year":"2002","journal-title":"IEEE Trans. Evol. Comput."},{"key":"ref_23","first-page":"32","article-title":"An Improved Parallel Biobjective Hybrid Real-Coded Genetic Algorithm with Clustering-Based Selection","volume":"24","author":"Akopov","year":"2024","journal-title":"Cybern. Inf. Technol."},{"doi-asserted-by":"crossref","unstructured":"Nguyen, T.L., and Nguyen, Q.A. (2025). A Multi-Objective PSO-GWO Approach for Smart Grid Reconfiguration with Renewable Energy and Electric Vehicles. Energies, 18.","key":"ref_24","DOI":"10.3390\/en18082020"},{"key":"ref_25","doi-asserted-by":"crossref","first-page":"73","DOI":"10.1109\/MCI.2017.2742868","article-title":"PlatEMO: A MATLAB Platform for Evolutionary Multi-Objective Optimization [Educational Forum]","volume":"12","author":"Tian","year":"2017","journal-title":"IEEE Comput. Intell. Mag."},{"key":"ref_26","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1145\/3594261.3594262","article-title":"An interview with Kalyanmoy Deb 2022 ACM fellow","volume":"16","author":"Deb","year":"2023","journal-title":"ACM SIGEVOlution"},{"key":"ref_27","doi-asserted-by":"crossref","first-page":"381","DOI":"10.1016\/S0168-874X(00)00054-8","article-title":"Optimising frame structures by different strategies of genetic algorithms","volume":"37","author":"Greiner","year":"2001","journal-title":"Finite Elem. Anal. Des."},{"unstructured":"Bathe, K.J. (2006). Finite Element Procedures, Prentice Hall.","key":"ref_28"},{"unstructured":"(1993). Eurocode 3: Design of Steel Structures (Standard No. EN 1993).","key":"ref_29"},{"unstructured":"Hern\u00e1ndez-Ib\u00e1\u00f1ez, S. (1990). Structural Optimum Design Methods. Colecci\u00f3n Seinor, Colegio de Ingenieros de Caminos, Canales y Puertos. (In Spanish).","key":"ref_30"},{"unstructured":"Kingma, D.P., and Ba, J. (2014). Adam: A method for stochastic optimization. arXiv.","key":"ref_31"},{"key":"ref_32","doi-asserted-by":"crossref","first-page":"2357","DOI":"10.1109\/TSP.2019.2904921","article-title":"Learning ReLU networks on linearly separable data: Algorithm, optimality, and generalization","volume":"67","author":"Wang","year":"2019","journal-title":"IEEE Trans. Signal Process."},{"key":"ref_33","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1145\/3453474","article-title":"The hypervolume indicator: Computational problems and algorithms","volume":"54","author":"Guerreiro","year":"2021","journal-title":"ACM Comput. Surv."},{"unstructured":"Python Software Foundation (2025, November 10). Python: Version 3.12.7 Documentation. Available online: https:\/\/docs.python.org\/3\/.","key":"ref_34"},{"key":"ref_35","doi-asserted-by":"crossref","first-page":"89497","DOI":"10.1109\/ACCESS.2020.2990567","article-title":"Pymoo: Multi-objective optimization in Python","volume":"8","author":"Blank","year":"2020","journal-title":"IEEE Access"},{"unstructured":"Abadi, M., Barham, P., Chen, J., Chen, Z., Davis, A., Dean, J., Devin, M., Ghemawat, S., Irving, G., and Isard, M. (2016). TensorFlow: A System for Large-Scale Machine Learning. arXiv.","key":"ref_36"},{"key":"ref_37","first-page":"2677","article-title":"An extension on \u2018statistical comparisons of classifiers over multiple data sets\u2019 for all pairwise comparisons","volume":"9","author":"Garcia","year":"2007","journal-title":"J. Mach. Learn. Res."},{"key":"ref_38","doi-asserted-by":"crossref","first-page":"483","DOI":"10.1137\/130932715","article-title":"A survey of projection-based model reduction methods for parametric dynamical systems","volume":"57","author":"Benner","year":"2015","journal-title":"SIAM Rev."},{"key":"ref_39","doi-asserted-by":"crossref","first-page":"163","DOI":"10.1007\/s00158-024-03866-z","article-title":"Development of a multi-fidelity optimisation strategy based on hybrid methods","volume":"67","author":"Bugeda","year":"2024","journal-title":"Struct. Multidiscip. Optim."}],"container-title":["Algorithms"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/1999-4893\/18\/12\/754\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,12,5]],"date-time":"2025-12-05T18:58:19Z","timestamp":1764961099000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/1999-4893\/18\/12\/754"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2025,11,28]]},"references-count":39,"journal-issue":{"issue":"12","published-online":{"date-parts":[[2025,12]]}},"alternative-id":["a18120754"],"URL":"https:\/\/doi.org\/10.3390\/a18120754","relation":{},"ISSN":["1999-4893"],"issn-type":[{"type":"electronic","value":"1999-4893"}],"subject":[],"published":{"date-parts":[[2025,11,28]]}}}