{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,3]],"date-time":"2026-04-03T20:05:19Z","timestamp":1775246719294,"version":"3.50.1"},"reference-count":51,"publisher":"Springer Science and Business Media LLC","issue":"3","license":[{"start":{"date-parts":[[2019,12,11]],"date-time":"2019-12-11T00:00:00Z","timestamp":1576022400000},"content-version":"tdm","delay-in-days":0,"URL":"http:\/\/www.springer.com\/tdm"},{"start":{"date-parts":[[2019,12,11]],"date-time":"2019-12-11T00:00:00Z","timestamp":1576022400000},"content-version":"vor","delay-in-days":0,"URL":"http:\/\/www.springer.com\/tdm"}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"short-container-title":["J Optim Theory Appl"],"published-print":{"date-parts":[[2020,12]]},"DOI":"10.1007\/s10957-019-01614-8","type":"journal-article","created":{"date-parts":[[2019,12,11]],"date-time":"2019-12-11T12:55:28Z","timestamp":1576068928000},"page":"630-653","update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":96,"title":["Machine-Learning Techniques for the Optimal Design of Acoustic Metamaterials"],"prefix":"10.1007","volume":"187","author":[{"given":"Andrea","family":"Bacigalupo","sequence":"first","affiliation":[]},{"ORCID":"https:\/\/orcid.org\/0000-0002-5427-4328","authenticated-orcid":false,"given":"Giorgio","family":"Gnecco","sequence":"additional","affiliation":[]},{"given":"Marco","family":"Lepidi","sequence":"additional","affiliation":[]},{"given":"Luigi","family":"Gambarotta","sequence":"additional","affiliation":[]}],"member":"297","published-online":{"date-parts":[[2019,12,11]]},"reference":[{"issue":"2121","key":"1614_CR1","first-page":"2495","volume":"466","author":"NA Fleck","year":"2010","unstructured":"Fleck, N.A., Deshpande, V.S., Ashby, M.F.: Micro-architectured materials: past, present and future. Proc. R. Soc. A Math. Phys. Eng. Sci. 466(2121), 2495\u20132516 (2010)","journal-title":"Proc. R. Soc. A Math. Phys. Eng. Sci."},{"key":"1614_CR2","doi-asserted-by":"crossref","first-page":"187","DOI":"10.1146\/annurev-matsci-070115-031624","volume":"46","author":"MF Schaedler","year":"2016","unstructured":"Schaedler, M.F., Carter, W.B.: Architected cellular materials. Annu. Rev. Mater. Res. 46, 187\u2013210 (2016)","journal-title":"Annu. Rev. Mater. Res."},{"issue":"37","key":"1614_CR3","doi-asserted-by":"crossref","first-page":"11502","DOI":"10.1073\/pnas.1509120112","volume":"112","author":"LR Meza","year":"2015","unstructured":"Meza, L.R., Zelhofer, A.J., Clarke, N., Mateos, A.J., Kochmann, D.M., Greer, J.R.: Resilient 3D hierarchical architected metamaterials. Proc. Natl. Acad. Sci. 112(37), 11502\u201311507 (2015)","journal-title":"Proc. Natl. Acad. Sci."},{"issue":"5485","key":"1614_CR4","doi-asserted-by":"crossref","first-page":"1734","DOI":"10.1126\/science.289.5485.1734","volume":"289","author":"Z Liu","year":"2000","unstructured":"Liu, Z., Zhang, X., Mao, Y., Zhu, Y.Y., Yang, Z., Chan, C.T., Sheng, P.: Locally resonant sonic materials. Science 289(5485), 1734\u20131736 (2000)","journal-title":"Science"},{"issue":"12","key":"1614_CR5","doi-asserted-by":"crossref","first-page":"34","DOI":"10.1016\/S1369-7021(09)70315-3","volume":"12","author":"MH Lu","year":"2009","unstructured":"Lu, M.H., Feng, L., Chen, Y.F.: Phononic crystals and acoustic metamaterials. Mater. Today 12(12), 34\u201342 (2009)","journal-title":"Mater. Today"},{"issue":"2","key":"1614_CR6","doi-asserted-by":"crossref","first-page":"e1501595","DOI":"10.1126\/sciadv.1501595","volume":"2","author":"G Ma","year":"2016","unstructured":"Ma, G., Sheng, P.: Acoustic metamaterials: from local resonances to broad horizons. Sci. Adv. 2(2), e1501595 (2016)","journal-title":"Sci. Adv."},{"key":"1614_CR7","volume-title":"Dynamics of lattice materials","year":"2017","unstructured":"Phani, A.S., Hussein, M.I. (eds.): Dynamics of lattice materials. Wiley, New York (2017)"},{"issue":"19","key":"1614_CR8","doi-asserted-by":"crossref","first-page":"194301","DOI":"10.1103\/PhysRevLett.102.194301","volume":"102","author":"S Zhang","year":"2009","unstructured":"Zhang, S., Yin, L., Fang, N.: Focusing ultrasound with an acoustic metamaterial network. Phys. Rev. Lett. 102(19), 194301 (2009)","journal-title":"Phys. Rev. Lett."},{"key":"1614_CR9","volume-title":"Acoustic metamaterials: negative refraction, imaging, lensing and cloaking","year":"2012","unstructured":"Craster, R.V., Guenneau, S. (eds.): Acoustic metamaterials: negative refraction, imaging, lensing and cloaking, vol. 166. Springer, Berlin (2012)"},{"key":"1614_CR10","doi-asserted-by":"crossref","first-page":"174303","DOI":"10.1103\/PhysRevB.87.174303","volume":"87","author":"D Bigoni","year":"2013","unstructured":"Bigoni, D., Guenneau, S., Movchan, A.B., Brun, M.: Elastic metamaterials with inertial locally resonant structures: application to lensing and localization. Phys. Rev. B 87, 174303 (2013)","journal-title":"Phys. Rev. B"},{"key":"1614_CR11","doi-asserted-by":"crossref","first-page":"8037","DOI":"10.1038\/ncomms9037","volume":"6","author":"M Moler\u00f3n","year":"2015","unstructured":"Moler\u00f3n, M., Daraio, C.: Acoustic metamaterial for subwavelength edge detection. Nat. Commun. 6, 8037 (2015)","journal-title":"Nat. Commun."},{"key":"1614_CR12","doi-asserted-by":"crossref","first-page":"126","DOI":"10.1016\/j.ijsolstr.2016.01.005","volume":"83","author":"A Bacigalupo","year":"2016","unstructured":"Bacigalupo, A., Gambarotta, L.: Simplified modelling of chiral lattice materials with local resonators. Int. J. Solids Struct. 83, 126\u2013141 (2016)","journal-title":"Int. J. Solids Struct."},{"issue":"6","key":"1614_CR13","doi-asserted-by":"crossref","first-page":"418","DOI":"10.1007\/s00158-004-0497-6","volume":"29","author":"AR Diaz","year":"2005","unstructured":"Diaz, A.R., Haddow, A.G., Ma, L.: Design of band-gap grid structures. Struct. Multidiscip. Optim. 29(6), 418\u2013431 (2005)","journal-title":"Struct. Multidiscip. Optim."},{"issue":"20","key":"1614_CR14","doi-asserted-by":"crossref","first-page":"4406","DOI":"10.1016\/j.jsv.2012.05.027","volume":"331","author":"H Meng","year":"2012","unstructured":"Meng, H., Wen, J., Zhao, H., Wen, X.: Optimization of locally resonant acoustic metamaterials on underwater sound absorption characteristics. J. Sound Vib. 331(20), 4406\u20134416 (2012)","journal-title":"J. Sound Vib."},{"issue":"5","key":"1614_CR15","doi-asserted-by":"crossref","first-page":"054009","DOI":"10.1088\/0964-1726\/25\/5\/054009","volume":"25","author":"A Bacigalupo","year":"2016","unstructured":"Bacigalupo, A., Lepidi, M., Gnecco, G., Gambarotta, L.: Optimal design of auxetic hexachiral metamaterials with local resonators. Smart Mater. Struct. 25(5), 054009 (2016)","journal-title":"Smart Mater. Struct."},{"issue":"5","key":"1614_CR16","doi-asserted-by":"crossref","first-page":"341","DOI":"10.1016\/j.compositesb.2016.09.062","volume":"115","author":"A Bacigalupo","year":"2017","unstructured":"Bacigalupo, A., Gnecco, G., Lepidi, M., Gambarotta, L.: Optimal design of low-frequency band gaps in anti-tetrachiral lattice meta-materials. Compos. B Eng. 115(5), 341\u2013359 (2017)","journal-title":"Compos. B Eng."},{"key":"1614_CR17","doi-asserted-by":"crossref","first-page":"1","DOI":"10.3389\/fmats.2019.00002","volume":"6","author":"A Bacigalupo","year":"2019","unstructured":"Bacigalupo, A., Lepidi, M., Gnecco, G., Vadal\u00e0, F., Gambarotta, L.: Optimal design of the band structure for beam lattice metamaterials. Front. Mater. 6, 1\u201314 (2019)","journal-title":"Front. Mater."},{"issue":"13","key":"1614_CR18","doi-asserted-by":"crossref","first-page":"20012027","DOI":"10.1007\/s11012-019-00981-w","volume":"54","author":"M Bruggi","year":"2019","unstructured":"Bruggi, M., Corigliano, A.: Optimal 2D auxetic micro-structures with band gap. Meccanica 54(13), 20012027 (2019)","journal-title":"Meccanica"},{"key":"1614_CR19","doi-asserted-by":"crossref","DOI":"10.1002\/0471787779","volume-title":"Nonlinear Programming: Theory and Algorithms","author":"MS Bazaraa","year":"2006","unstructured":"Bazaraa, M.S., Sherali, H.D., Shetty, C.M.: Nonlinear Programming: Theory and Algorithms. Wiley, New York (2006)"},{"key":"1614_CR20","volume-title":"Surrogate-Based Modeling and Optimization: Applications in Engineering","year":"2013","unstructured":"Koziel, S., Leifsson, L. (eds.): Surrogate-Based Modeling and Optimization: Applications in Engineering. Springer, Berlin (2013)"},{"key":"1614_CR21","doi-asserted-by":"crossref","first-page":"349","DOI":"10.1137\/120902434","volume":"55","author":"SM Wild","year":"2013","unstructured":"Wild, S.M., Shoemaker, C.: Global convergence of radial basis function trust-region algorithms for derivative-free optimization. SIAM Rev. 55, 349\u2013371 (2013)","journal-title":"SIAM Rev."},{"key":"1614_CR22","doi-asserted-by":"crossref","unstructured":"Bacigalupo, A., Gnecco, G., Lepidi, M., Gambarotta, L.: Design of acoustic metamaterials through nonlinear programming. In: 2nd International Workshop on Optimization, Machine Learning and Big Data (MOD 2016). Lecture Notes in Computer Science, vol. 10122, pp. 170\u2013181 (2016)","DOI":"10.1007\/978-3-319-51469-7_14"},{"key":"1614_CR23","doi-asserted-by":"crossref","unstructured":"Bacigalupo, A., Gnecco, G.: Metamaterial filter design via surrogate optimization. In: International Conference on Metamaterials and Nanophotonics (METANANO 2018). Journal of Physics: Conference Series, vol. 1092, pp. 1\u20134 (2018)","DOI":"10.1088\/1742-6596\/1092\/1\/012043"},{"key":"1614_CR24","doi-asserted-by":"crossref","first-page":"179","DOI":"10.1016\/j.jmps.2014.07.004","volume":"71","author":"AO Krushynska","year":"2014","unstructured":"Krushynska, A.O., Kouznetsova, V.G., Geers, M.G.D.: Towards optimal design of locally resonant acoustic metamaterials. J. Mech. Phys. Solids 71, 179\u2013196 (2014)","journal-title":"J. Mech. Phys. Solids"},{"key":"1614_CR25","doi-asserted-by":"crossref","first-page":"186","DOI":"10.1016\/j.ijsolstr.2017.12.014","volume":"136","author":"M Lepidi","year":"2018","unstructured":"Lepidi, M., Bacigalupo, A.: Multi-parametric sensitivity analysis of the band structure for tetrachiral acoustic metamaterials. Int. J. Solids Struct. 136, 186\u2013202 (2018)","journal-title":"Int. J. Solids Struct."},{"issue":"1","key":"1614_CR26","doi-asserted-by":"crossref","first-page":"2262","DOI":"10.1038\/s41598-018-19963-1","volume":"8","author":"L D\u2019Alessandro","year":"2018","unstructured":"D\u2019Alessandro, L., Zega, V., Ardito, R., Corigliano, A.: 3D auxetic single material periodic structure with ultra-wide tunable bandgap. Sci. Rep. 8(1), 2262 (2018)","journal-title":"Sci. Rep."},{"key":"1614_CR27","first-page":"1","volume":"5","author":"J Schmidt","year":"2019","unstructured":"Schmidt, J., Marques, M.R.G., Botti, S., Marques, M.A.L.: Recent advances and applications of machine learning in solid-state materials science. Nat. Comput. Mater. 5, 1\u201336 (2019)","journal-title":"Nat. Comput. Mater."},{"key":"1614_CR28","doi-asserted-by":"crossref","DOI":"10.1142\/6437","volume-title":"Meshfree Approximation Methods with MATLAB","author":"GE Fasshauer","year":"2007","unstructured":"Fasshauer, G.E.: Meshfree Approximation Methods with MATLAB. World Scientific, Singapore (2007)"},{"key":"1614_CR29","doi-asserted-by":"crossref","DOI":"10.1137\/1.9781611970081","volume-title":"Random Number Generation and Quasi-Monte Carlo Methods","author":"H Niederreiter","year":"1992","unstructured":"Niederreiter, H.: Random Number Generation and Quasi-Monte Carlo Methods. SIAM, Philadelphia (1992)"},{"key":"1614_CR30","doi-asserted-by":"crossref","first-page":"193","DOI":"10.1023\/A:1018975909870","volume":"11","author":"S Rippa","year":"1999","unstructured":"Rippa, S.: An algorithm for selecting a good value for the parameter c in radial basis function interpolation. Adv. Comput. Math. 11, 193\u2013210 (1999)","journal-title":"Adv. Comput. Math."},{"key":"1614_CR31","doi-asserted-by":"crossref","DOI":"10.1017\/CBO9780511801389","volume-title":"An Introduction to Support Vector Machines and Other Kernel-Based Methods","author":"S Cristianini","year":"2000","unstructured":"Cristianini, S., Shawe-Taylor, J.: An Introduction to Support Vector Machines and Other Kernel-Based Methods. Cambridge University Press, Cambridge (2000)"},{"key":"1614_CR32","first-page":"632","volume-title":"Advances in Structural and Multidisciplinary Optimization","author":"C Lin","year":"2017","unstructured":"Lin, C., Lee, Y.H., Schuh, J.K., Ewoldt, R.H., Allison, J.T.: Efficient optimal surface texture design using linearization. Advances in Structural and Multidisciplinary Optimization, pp. 632\u2013647. Springer, New York (2017)"},{"key":"1614_CR33","doi-asserted-by":"crossref","first-page":"359","DOI":"10.1002\/nme.1620240207","volume":"24","author":"K Svanberg","year":"1987","unstructured":"Svanberg, K.: The method of moving asymptotes\u2014a new method for structural optimization. Int. J. Numer. Meth. Eng. 24, 359\u2013373 (1987)","journal-title":"Int. J. Numer. Meth. Eng."},{"key":"1614_CR34","doi-asserted-by":"crossref","first-page":"555","DOI":"10.1137\/S1052623499362822","volume":"12","author":"K Svanberg","year":"2002","unstructured":"Svanberg, K.: A class of globally convergent optimization methods based on conservative convex separable approximations. SIAM J. Optim. 12, 555\u2013573 (2002)","journal-title":"SIAM J. Optim."},{"key":"1614_CR35","doi-asserted-by":"crossref","first-page":"667","DOI":"10.1007\/s00158-013-1005-7","volume":"49","author":"P Wei","year":"2014","unstructured":"Wei, P., Ma, H., Wang, M.Y.: The stiffness spreading method for layout optimization of truss structures. Struct. Multidiscip. Optim. 49, 667\u2013682 (2014)","journal-title":"Struct. Multidiscip. Optim."},{"key":"1614_CR36","doi-asserted-by":"crossref","first-page":"86","DOI":"10.1016\/0041-5553(67)90144-9","volume":"7","author":"IM Sobol","year":"1967","unstructured":"Sobol, I.M.: The distribution of points in a cube and the approximate evaluation of integrals. USSR Comput. Math. Math. Phys. 7, 86\u2013112 (1967)","journal-title":"USSR Comput. Math. Math. Phys."},{"key":"1614_CR37","doi-asserted-by":"crossref","DOI":"10.7551\/mitpress\/3927.001.0001","volume-title":"An Introduction to Genetic Algorithms","author":"M Mitchell","year":"1996","unstructured":"Mitchell, M.: An Introduction to Genetic Algorithms. MIT Press, Cambridge (1996)"},{"key":"1614_CR38","doi-asserted-by":"crossref","first-page":"123","DOI":"10.1007\/s10898-014-0184-0","volume":"60","author":"J M\u00fcller","year":"2014","unstructured":"M\u00fcller, J., Shoemaker, C.A.: Influence of ensemble surrogate models and sampling strategy on the solution quality of algorithms for computationally expensive black-box global optimization problems. J. Global Optim. 60, 123\u2013144 (2014)","journal-title":"J. Global Optim."},{"key":"1614_CR39","doi-asserted-by":"crossref","first-page":"2203","DOI":"10.1162\/neco_a_00976","volume":"29","author":"G Gnecco","year":"2017","unstructured":"Gnecco, G., Bemporad, A., Gori, M., Sanguineti, M.: LQG online learning. Neural Comput. 29, 2203\u20132291 (2017)","journal-title":"Neural Comput."},{"key":"1614_CR40","doi-asserted-by":"crossref","unstructured":"Zoppoli, R., Sanguineti, M., Gnecco, G., Parisini, T.: Neural Approximations for Optimal Control and Decision. Springer (2020) (forthcoming)","DOI":"10.1007\/978-3-030-29693-3"},{"key":"1614_CR41","doi-asserted-by":"crossref","first-page":"380","DOI":"10.1007\/s10957-012-0118-2","volume":"156","author":"M Gaggero","year":"2013","unstructured":"Gaggero, M., Gnecco, G., Sanguineti, M.: Dynamic programming and value-function approximation in sequential decision problems: error analysis and numerical results. J. Optim. Theory Appl. 156, 380\u2013416 (2013)","journal-title":"J. Optim. Theory Appl."},{"key":"1614_CR42","unstructured":"Girosi, F.: Approximation error bounds that use VC-bounds. In: Proceedings of the International Conference on Artificial Neural Networks, pp. 295\u2013302 (1995)"},{"key":"1614_CR43","doi-asserted-by":"crossref","first-page":"764","DOI":"10.1007\/s10957-010-9680-7","volume":"146","author":"G Gnecco","year":"2010","unstructured":"Gnecco, G., Sanguineti, M.: Suboptimal solutions to dynamic optimization problems via approximations of the policy function. J. Optim. Theory Appl. 146, 764\u2013794 (2010)","journal-title":"J. Optim. Theory Appl."},{"key":"1614_CR44","unstructured":"Barron, A.R.: Neural net approximation. In: Proceedings of the 7th Yale Workshop on Adaptive and Learning Systems, pp. 69\u201372 (1992)"},{"key":"1614_CR45","doi-asserted-by":"crossref","first-page":"33","DOI":"10.1007\/s10957-008-9471-6","volume":"140","author":"S Giulini","year":"2019","unstructured":"Giulini, S., Sanguineti, M.: Approximation schemes for functional optimization problems. J. Optim. Theory Appl. 140, 33\u201354 (2019)","journal-title":"J. Optim. Theory Appl."},{"key":"1614_CR46","doi-asserted-by":"crossref","first-page":"461","DOI":"10.1137\/S1052623403426507","volume":"15","author":"V K\u016frkov\u00e1","year":"2005","unstructured":"K\u016frkov\u00e1, V., Sanguineti, M.: Error estimates for approximate optimization by the extended Ritz method. SIAM J. Optim. 15, 461\u2013487 (2005)","journal-title":"SIAM J. Optim."},{"key":"1614_CR47","doi-asserted-by":"crossref","first-page":"403","DOI":"10.1023\/A:1013662124879","volume":"112","author":"R Zoppoli","year":"2002","unstructured":"Zoppoli, R., Sanguineti, M., Parisini, T.: Approximating networks and extended Ritz method for the solution of functional optimization problems. J. Optim. Theory Appl. 112, 403\u2013439 (2002)","journal-title":"J. Optim. Theory Appl."},{"key":"1614_CR48","doi-asserted-by":"crossref","first-page":"925","DOI":"10.1007\/s10957-018-1396-0","volume":"180","author":"AM Schweidtmann","year":"2019","unstructured":"Schweidtmann, A.M., Mitsos, A.: Deterministic global optimization with artificial neural networks embedded. J. Optim. Theory Appl. 180, 925\u2013948 (2019)","journal-title":"J. Optim. Theory Appl."},{"key":"1614_CR49","doi-asserted-by":"crossref","unstructured":"Gnecco, G.: An algorithm for curve identification in the presence of curve intersections. In: Mathematical Problems in Engineering, pp. 1\u20137 (2018)","DOI":"10.1155\/2018\/7243691"},{"key":"1614_CR50","doi-asserted-by":"crossref","first-page":"242","DOI":"10.1016\/j.laa.2009.08.001","volume":"432","author":"Y Nakatsukasa","year":"2010","unstructured":"Nakatsukasa, Y.: Absolute and relative Weyl theorems for generalized eigenvalue problems. Linear Algebra Appl. 432, 242\u2013248 (2010)","journal-title":"Linear Algebra Appl."},{"key":"1614_CR51","doi-asserted-by":"publisher","DOI":"10.1007\/s11590-019-01486-x","author":"G Gnecco","year":"2019","unstructured":"Gnecco, G., Nutarelli, F.: On the trade-off between number of examples and precision of supervision in machine learning problems. Optim. Lett. (2019). https:\/\/doi.org\/10.1007\/s11590-019-01486-x","journal-title":"Optim. Lett."}],"container-title":["Journal of Optimization Theory and Applications"],"original-title":[],"language":"en","link":[{"URL":"http:\/\/link.springer.com\/content\/pdf\/10.1007\/s10957-019-01614-8.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"http:\/\/link.springer.com\/article\/10.1007\/s10957-019-01614-8\/fulltext.html","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"http:\/\/link.springer.com\/content\/pdf\/10.1007\/s10957-019-01614-8.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2024,7,28]],"date-time":"2024-07-28T12:14:52Z","timestamp":1722168892000},"score":1,"resource":{"primary":{"URL":"http:\/\/link.springer.com\/10.1007\/s10957-019-01614-8"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2019,12,11]]},"references-count":51,"journal-issue":{"issue":"3","published-print":{"date-parts":[[2020,12]]}},"alternative-id":["1614"],"URL":"https:\/\/doi.org\/10.1007\/s10957-019-01614-8","relation":{},"ISSN":["0022-3239","1573-2878"],"issn-type":[{"value":"0022-3239","type":"print"},{"value":"1573-2878","type":"electronic"}],"subject":[],"published":{"date-parts":[[2019,12,11]]},"assertion":[{"value":"14 June 2019","order":1,"name":"received","label":"Received","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"13 November 2019","order":2,"name":"accepted","label":"Accepted","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"11 December 2019","order":3,"name":"first_online","label":"First Online","group":{"name":"ArticleHistory","label":"Article History"}},{"order":1,"name":"Ethics","group":{"name":"EthicsHeading","label":"Compliance with Ethical Standards"}},{"value":"The authors declare that they have no conflict of interest.","order":2,"name":"Ethics","group":{"name":"EthicsHeading","label":"Conflict of interest"}}]}}