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Designing phononic crystal with anticipated band gap through a deep learning based data-driven method. (English) Zbl 1442.74135
Summary: Phononic crystal is a type of artificial heterogeneous material constituted by a periodic repetition of cells. This characteristic provides a possible solution to the accurate manipulation of acoustic and elastic waves. For this reason, phononic crystal is of application potentials in vibration and noise reduction, filtering, acoustic lens, acoustic imaging, and acoustic stealth, etc. It is thus of significance in the fields of information, communication, and medical applications. To design phononic crystal with anticipated manipulation characteristic has become a research hotspot in recent years. However, how to accurately manipulate acoustic and mechanical wave is still a major challenge for existing designing approaches. Assisted by image-based finite element analysis and deep learning, a data-driven approach is proposed in this study for designing phononic crystals. An auto-encoder is trained to extract the topological features from sample images. Finite element analysis is employed to study the band gaps of samples. A multi-layer perceptron is trained to establish the inherent relation between band gaps and topological features. The trained models are ultimately employed to design phononic crystals with anticipated band gaps. Not limited to this material, the proposed method could be further extended to design various structured mechanical materials with specific functionalities.
74M05 Control, switches and devices (“smart materials”) in solid mechanics
65Z05 Applications to the sciences
darch; Matlab; top88.m; top.m
Full Text: DOI
[1] Lee, J. H.; Singer, J. P.; Thomas, E. L., Micro-/nanostructured mechanical metamaterials, Adv. Mater., 24, 36, 4782-4810 (2012)
[2] Kushwaha, M. S.; Halevi, P.; Dobrzynski, L.; Djafari-Rouhani, B., Acoustic band structure of periodic elastic composites, Phys. Rev. Lett., 71, 13, 2022 (1993)
[3] Martínez-Sala, R., Sound attenuation by sculpture, Nature, 378, 241 (1995)
[4] Page, J. H.; Sheng, P.; Schriemer, H. P.; Jones, I.; Jing, X.; Weitz, D. A., Group velocity in strongly scattering media, Science, 271, 5249, 634-637 (1996)
[5] Pennec, Y.; Djafari-Rouhani, B., Fundamental Properties of Phononic Crystal, Phononic Crystals, 23-50 (2016), Springer
[6] Schriemer, H. P.; Cowan, M. L.; Page, J. H.; Sheng, P.; Liu, Z.; Weitz, D. A., Energy velocity of diffusing waves in strongly scattering media, Phys. Rev. Lett., 79, 17, 3166 (1997)
[7] Sigalas, M. M.; Economou, E. N., Elastic and acoustic wave band structure, J. Sound Vib., 158, 377-382 (1992)
[8] Srivastava, G. P., The Physics of Phonons (1990), CRC press
[9] Zheng, L.-Y., Granular Monolayers: Wave Dynamics and Topological Properties (2017), Université du Maine
[10] Vasseur, J.; Matar, O. B.; Robillard, J.; Hladky-Hennion, A.-C.; Deymier, P. A., Band structures tunability of bulk 2D phononic crystals made of magneto-elastic materials, AIP Adv., 1, 4, Article 041904 pp. (2011)
[11] Xie, Y.; Wang, W.; Chen, H.; Konneker, A.; Popa, B.-I.; Cummer, S. A., Wavefront modulation and subwavelength diffractive acoustics with an acoustic metasurface, Nat. Commun., 5, 5553 (2014)
[12] Pennec, Y.; Vasseur, J. O.; Djafari-Rouhani, B.; Dobrzyński, L.; Deymier, P. A., Two-dimensional phononic crystals: Examples and applications, Surf. Sci. Rep., 65, 8, 229-291 (2010)
[13] Liu, Z.; Zhang, X.; Mao, Y.; Zhu, Y.; Yang, Z.; Chan, C. T.; Sheng, P., Locally resonant sonic materials, Science, 289, 5485, 1734-1736 (2000)
[14] Hu, X.; Shen, Y.; Liu, X.; Fu, R.; Zi, J., Superlensing effect in liquid surface waves, Phys. Rev. E, 69, 3, Article 030201 pp. (2004)
[15] Cervera, F.; Sanchis, L.; Sánchez-Pérez, J.; Martinez-Sala, R.; Rubio, C.; Meseguer, F.; López, C.; Caballero, D.; Sánchez-Dehesa, J., Refractive acoustic devices for airborne sound, Phys. Rev. Lett., 88, 2, Article 023902 pp. (2001)
[16] Kushwaha, M. S.; Halevi, P.; Dobrzynski, L.; Djafari-Rouhani, B., Acoustic band structure of periodic elastic composites, Phys. Rev. Lett., 71, 13, 2022 (1993)
[17] Kushwaha, M. S.; Halevi, P.; Martinez, G.; Dobrzynski, L.; Djafari-Rouhani, B., Theory of acoustic band structure of periodic elastic composites, Phys. Rev. B, 49, 4, 2313 (1994)
[18] Sigalas, M.; Economou, E. N., Band structure of elastic waves in two dimensional systems, Solid State Commun., 86, 3, 141-143 (1993)
[19] Vasseur, J.; Djafari-Rouhani, B.; Dobrzynski, L.; Kushwaha, M.; Halevi, P., Complete acoustic band gaps in periodic fibre reinforced composite materials: the carbon/epoxy composite and some metallic systems, J. Phys.: Condens. Matter., 6, 42, 8759 (1994)
[20] Sigalas, M.; Kushwaha, M. S.; Economou, E. N.; Kafesaki, M.; Psarobas, I. E.; Steurer, W., Classical vibrational modes in phononic lattices: theory and experiment, Z. für Kristallographie-Crystalline Mater., 220, 9-10, 765-809 (2005)
[21] Wang, P.; Shim, J.; Bertoldi, K., Effects of geometric and material nonlinearities on tunable band gaps and low-frequency directionality of phononic crystals, Phys. Rev. B, 88, 1, Article 014304 pp. (2013)
[22] Huang, H.; Sun, C., Continuum modeling of a composite material with internal resonators, Mech. Mater., 46, 1-10 (2012)
[23] Krödel, S.; Thomé, N.; Daraio, C., Wide band-gap seismic metastructures, Extreme Mech. Lett., 4, 111-117 (2015)
[24] Nanthakumar, S. S.; Zhuang, X.; Park, H. S.; Nguyen, C.; Chen, Y.; Rabczuk, T., Inverse design of quantum spin hall-based phononic topological insulators, J. Mech. Phys. Solids, 125, 550-571 (2019)
[25] Nguyen, B. H.; Zhuang, X.; Park, H. S.; Rabczuk, T., Tunable topological bandgaps and frequencies in a pre-stressed soft phononic crystal, J. Appl. Phys., 125, 9 (2019)
[26] Halkjær, S.; Sigmund, O.; Jensen, J. S., Inverse design of phononic crystals by topology optimization, Z. Kristallogr.-Cryst. Mater., 220, 9-10, 895-905 (2005)
[27] Sigmund, O.; Jensen, J. S., Systematic design of phononic band-gap materials and structures by topology optimization, Phil. Trans. R. Soc. Lond. A Math. Phys. Eng. Sci., 361, 1806, 1001-1019 (2003) · Zbl 1067.74053
[28] Liu, K.; Tovar, A., An efficient 3D topology optimization code written in Matlab, Struct. Multidiscip. Optim., 50, 6, 1175-1196 (2014)
[29] Sigmund, O.; Maute, K., Topology optimization approaches, Struct. Multidiscip. Optim., 48, 6, 1031-1055 (2013)
[30] Bendsøe, M. P., Optimal shape design as a material distribution problem, Struct. Optim., 1, 4, 193-202 (1989)
[31] Rozvany, G. I.; Zhou, M.; Birker, T., Generalized shape optimization without homogenization, Struct. Optim., 4, 3-4, 250-252 (1992)
[32] Mlejnek, H., Some aspects of the genesis of structures, Struct. Optim., 5, 1-2, 64-69 (1992)
[33] Sigmund, O., A 99 line topology optimization code written in Matlab, Struct. Multidiscip. Optim., 21, 2, 120-127 (2001)
[34] Andreassen, E.; Clausen, A.; Schevenels, M.; Lazarov, B. S.; Sigmund, O., Efficient topology optimization in MATLAB using 88 lines of code, Struct. Multidiscip. Optim., 43, 1, 1-16 (2011) · Zbl 1274.74310
[35] Allaire, G.; Jouve, F.; Toader, A.-M., A level-set method for shape optimization, C. R. Math., 334, 12, 1125-1130 (2002) · Zbl 1115.49306
[36] Allaire, G.; Jouve, F.; Toader, A.-M., Structural optimization using sensitivity analysis and a level-set method, J. Comput. Phys., 194, 1, 363-393 (2004) · Zbl 1136.74368
[37] Wang, M. Y.; Wang, X.; Guo, D., A level set method for structural topology optimization, Comput. Methods Appl. Mech. Engrg., 192, 1-2, 227-246 (2003) · Zbl 1083.74573
[38] Bourdin, B.; Chambolle, A., Design-dependent loads in topology optimization, ESAIM Control Optim. Calc. Var., 9, 19-48 (2003) · Zbl 1066.49029
[39] Takezawa, A.; Nishiwaki, S.; Kitamura, M., Shape and topology optimization based on the phase field method and sensitivity analysis, J. Comput. Phys., 229, 7, 2697-2718 (2010) · Zbl 1185.65109
[40] Burger, M.; Stainko, R., Phase-field relaxation of topology optimization with local stress constraints, SIAM J. Control Optim., 45, 4, 1447-1466 (2006) · Zbl 1116.74053
[41] Sokołowski, J.; Żochowski, A., Topological derivative in shape optimization, (Encyclopedia of Optimization (2001), Springer), 2625-2626
[42] Norato, J. A.; Bendsøe, M. P.; Haber, R. B.; Tortorelli, D. A., A topological derivative method for topology optimization, Struct. Multidiscip. Optim., 33, 4-5, 375-386 (2007) · Zbl 1245.74074
[43] Xie, Y. M.; Steven, G. P., A simple evolutionary procedure for structural optimization, Comput. Struct., 49, 5, 885-896 (1993)
[44] Mattheck, C.; Burkhardt, S., A new method of structural shape optimization based on biological growth, Int. J. Fatigue, 12, 3, 185-190 (1990)
[45] Dong, H.-W.; Su, X.-X.; Wang, Y.-S., Multi-objective optimization of two-dimensional porous phononic crystals, J. Phys. D: Appl. Phys., 47, 15 (2014)
[46] Dong, H.-W.; Su, X.-X.; Wang, Y.-S.; Zhang, C., Topological optimization of two-dimensional phononic crystals based on the finite element method and genetic algorithm, Struct. Multidiscip. Optim., 50, 4, 593-604 (2014)
[47] Dong, H.-W.; Su, X.-X.; Wang, Y.-S.; Zhang, C., Topology optimization of two-dimensional asymmetrical phononic crystals, Phys. Lett. A, 378, 4, 434-441 (2014)
[48] Y.f. Li, H.-W.; Huang, X.; Meng, F.; Zhou, S., Evolutionary topological design for phononic band gap crystals, Struct. Multidiscip. Optim., 54, 3, 595-617 (2016)
[49] Rupp, C. J.; Evgrafov, A.; Maute, K.; Dunn, M. L., Design of phononic materials/structures for surface wave devices using topology optimization, Struct. Multidiscip. Optim., 34, 2, 111-121 (2006) · Zbl 1273.74405
[50] Zhang, X.; He, J.; Takezawa, A.; Kang, Z., Robust topology optimization of phononic crystals with random field uncertainty, Internat. J. Numer. Methods Engrg., 115, 9, 1154-1173 (2018)
[51] Bessa, M.; Bostanabad, R.; Liu, Z.; Hu, A.; Apley, D. W.; Brinson, C.; Chen, W.; Liu, W. K., A framework for data-driven analysis of materials under uncertainty: Countering the curse of dimensionality, Comput. Methods Appl. Mech. Engrg., 320, 633-667 (2017)
[52] McCulloch, W. S.; Pitts, W., A logical calculus of the ideas immanent in nervous activity, Bull. Math. Biophys., 5, 4, 115-133 (1943) · Zbl 0063.03860
[53] Hinton, G. E.; Osindero, S.; Teh, Y.-W., A fast learning algorithm for deep belief nets, Neural Comput., 18, 7, 1527-1554 (2006) · Zbl 1106.68094
[54] Hinton, G. E., Learning multiple layers of representation, Trends Cogn. Sci., 11, 10, 428-434 (2007)
[55] LeCun, Y.; Bengio, Y.; Hinton, G., Deep learning, Nature, 521, 7553, 436 (2015)
[56] Schmidhuber, J., Deep learning in neural networks: An overview, Neural Netw., 61, 85-117 (2015)
[57] Goodfellow, I.; Bengio, Y.; Courville, A.; Bengio, Y., Deep Learning (2016), MIT press Cambridge · Zbl 1373.68009
[58] Raccuglia, P.; Elbert, K. C.; Adler, P. D.; Falk, C.; Wenny, M. B.; Mollo, A.; Zeller, M.; Friedler, S. A.; Schrier, J.; Norquist, A. J., Machine-learning-assisted materials discovery using failed experiments, Nature, 533, 7601, 73 (2016)
[59] Sanchez-Lengeling, B.; Aspuru-Guzik, A., Inverse molecular design using machine learning: Generative models for matter engineering, Science, 361, 6400, 360-365 (2018)
[60] Silver, D.; Huang, A.; Maddison, C. J.; Guez, A.; Sifre, L.; Van, d. D.G.; Schrittwieser, J.; Antonoglou, I.; Panneershelvam, V.; Lanctot, M., Mastering the game of Go with deep neural networks and tree search, Nature, 529, 7587, 484-489 (2016)
[61] Le Cun, Y., (Learning Process in an Asymmetric Threshold Network, Disordered Systems and Biological Organization (1986), Springer), 233-240 · Zbl 1356.82026
[62] Rumelhart, D. E.; Hinton, G. E.; Williams, R. J., Learning representations by back-propagating errors, Nature, 323, 6088, 533 (1986) · Zbl 1369.68284
[63] Werbos, P., Beyond Regression: New Tools for Prediction and Analysis in the Behavioral Sciences (1974), Harvard University, (Ph. D. dissertation)
[64] Furukawa, T.; Yagawa, G., Implicit constitutive modelling for viscoplasticity using neural networks, Internat. J. Numer. Methods Engrg., 43, 2, 195-219 (1998) · Zbl 0926.74020
[65] Ghaboussi, J.; Pecknold, D. A.; Zhang, M.; Haj-Ali, R. M., Autoprogressive training of neural network constitutive models, Internat. J. Numer. Methods Engrg., 42, 1, 105-126 (1998) · Zbl 0915.73075
[66] Hashash, Y.; Jung, S.; Ghaboussi, J., Numerical implementation of a neural network based material model in finite element analysis, Internat. J. Numer. Methods Engrg., 59, 7, 989-1005 (2004) · Zbl 1065.74609
[67] Ji, G.; Li, F.; Li, Q.; Li, H.; Li, Z., A comparative study on Arrhenius-type constitutive model and artificial neural network model to predict high-temperature deformation behaviour in Aermet100 steel, Mater. Sci. Eng. A, 528, 13-14, 4774-4782 (2011)
[68] Jung, S.; Ghaboussi, J., Neural network constitutive model for rate-dependent materials, Comput. Struct., 84, 15-16, 955-963 (2006)
[69] Sun, Y.; Zeng, W.; Zhao, Y.; Qi, Y.; Ma, X.; Han, Y., Development of constitutive relationship model of Ti600 alloy using artificial neural network, Comput. Mater. Sci., 48, 3, 686-691 (2010)
[70] Guo, H.; Zhuang, X.; Rabczuk, T., A deep collocation method for the bending analysis of kirchhoff plate, Comput. Mater. Continua, 58, 2, 433-456 (2019)
[71] Beigzadeh, R.; Rahimi, M., Prediction of heat transfer and flow characteristics in helically coiled tubes using artificial neural networks, Int. Commun. Heat Mass Transfer, 39, 8, 1279-1285 (2012)
[72] Butz, T.; Von Stryk, O., Modelling and simulation of electro-and magnetorheological fluid dampers, ZAMM-J. Appl. Math. Mech. / Z. Angew. Math. Mech. Appl. Math. Mech., 82, 1, 3-20 (2002) · Zbl 1045.76055
[73] Faller, W. E.; Schreck, S. J., Unsteady fluid mechanics applications of neural networks, J. Aircr., 34, 1, 48-55 (1997)
[74] Mi, Y.; Ishii, M.; Tsoukalas, L., Flow regime identification methodology with neural networks and two-phase flow models, Nucl. Eng. Des., 204, 1-3, 87-100 (2001)
[75] Wang, D.; Liao, W., Modeling and control of magnetorheological fluid dampers using neural networks, Smart Mater. Struct., 14, 1, 111 (2004)
[76] Yuhong, Z.; Wenxin, H., Application of artificial neural network to predict the friction factor of open channel flow, Commun. Nonlinear Sci. Numer. Simul., 14, 5, 2373-2378 (2009)
[77] Cang, R.; Li, H.; Yao, H.; Jiao, Y.; Ren, Y., Improving direct physical properties prediction of heterogeneous materials from imaging data via convolutional neural network and a morphology-aware generative model, Comput. Mater. Sci., 150, 212-221 (2018)
[78] Koker, R.; Altinkok, N.; Demir, A., Neural network based prediction of mechanical properties of particulate reinforced metal matrix composites using various training algorithms, Mater. Des., 28, 2, 616-627 (2007)
[79] Kondo, R.; Yamakawa, S.; Masuoka, Y.; Tajima, S.; Asahi, R., Microstructure recognition using convolutional neural networks for prediction of ionic conductivity in ceramics, Acta Mater., 141, 29-38 (2017)
[80] Liu, Z.; Bessa, M.; Liu, W. K., Self-consistent clustering analysis: an efficient multi-scale scheme for inelastic heterogeneous materials, Comput. Methods Appl. Mech. Engrg., 306, 319-341 (2016)
[81] Li, X.; Liu, Z.; Cui, S.; Luo, C.; Li, C.; Zhuang, Z., Engineering, Predicting the effective mechanical property of heterogeneous materials by image based modeling and deep learning, Comput. Methods Appl. Mech. Eng., 347, 735-753 (2019)
[82] Dissanayake, M.; Phan-Thien, N., Neural-network-based approximations for solving partial differential equations, Commun. Numer. Methods Eng., 10, 3, 195-201 (1994) · Zbl 0802.65102
[83] Lagaris, I. E.; Likas, A.; Fotiadis, D. I., Artificial neural networks for solving ordinary and partial differential equations, IEEE Trans. Neural Netw., 9, 5, 987-1000 (1998)
[84] Ramuhalli, P.; Udpa, L.; Udpa, S. S., Finite-element neural networks for solving differential equations, IEEE Trans. Neural Netw., 16, 6, 1381-1392 (2005)
[85] Takeuchi, J.; Kosugi, Y., Neural network representation of finite element method, Neural Netw., 7, 2, 389-395 (1994)
[86] Anitescu, C.; Atroshchenko, E.; Alajlan, N.; Rabczuk, T., Artificial neural network methods for the solution of second order boundary value problems, Comput. Mater. Continua, 59, 1, 345-359 (2019)
[87] Goodfellow, I. J.; Pouget-Abadie, J.; Mirza, M.; Xu, B.; Warde-Farley, D.; Ozair, S.; Courville, A.; Bengio, Y., Generative adversarial networks, Adv. Neural Inf. Process. Syst., 3, 2672-2680 (2014)
[88] Kingma, D. P.; Welling, M., Auto-encoding variational bayes (2013), arXiv preprint arXiv:1312.6114
[89] Makhzani, A.; Shlens, J.; Jaitly, N.; Goodfellow, I.; Frey, B., Adversarial autoencoders (2015), arXiv preprint arXiv:1511.05644
[90] Cang, R.; Xu, Y.; Chen, S.; Liu, Y.; Jiao, Y.; Ren, M. Y., Microstructure representation and reconstruction of heterogeneous materials via deep belief network for computational material design, J. Mech. Des., 139, 7, Article 071404 pp. (2017)
[91] Li, X.; Zhang, Y.; Zhao, H.; Burkhart, C.; Brinson, L. C.; Chen, W., A transfer learning approach for microstructure reconstruction and structure-property predictions (2018), arXiv preprint arXiv:1805.02784
[92] Yang, Z.; Li, X.; Brinson, L. C.; Choudhary, A. N.; Chen, W.; Agrawal, A., Microstructural materials design via deep adversarial learning methodology (2018), arXiv preprint arXiv:1805.02791
[93] Chen, H.; Engkvist, O.; Wang, Y.; Olivecrona, M.; Blaschke, T., The rise of deep learning in drug discovery, Drug Discov. Today, 23, 6, 1241-1250 (2018)
[94] Kadurin, A.; Aliper, A.; Kazennov, A.; Mamoshina, P.; Vanhaelen, Q.; Khrabrov, K.; Zhavoronkov, A., The cornucopia of meaningful leads: Applying deep adversarial autoencoders for new molecule development in oncology, Oncotarget, 8, 7, 10883 (2017)
[95] Segler, M. H.; Kogej, T.; Tyrchan, C.; Waller, M. P., Generating focused molecule libraries for drug discovery with recurrent neural networks, ACS Cent. Sci., 4, 1, 120-131 (2017)
[96] Eraslan, G.; Simon, L. M.; Mircea, M.; Mueller, N. S.; Theis, F. J., Single-cell RNA-seq denoising using a deep count autoencoder, Nat. Commun., 10, 1, 390 (2019)
[97] Grønbech, C. H.; Vording, M. F.; Timshel, P. N.; Sønderby, C. K.; Pers, T. H.; Winther, O., ScVAE: Variational auto-encoders for single-cell gene expression data, biorxiv, 318295 (2018)
[98] Riesselman, A. J.; Ingraham, J. B.; Marks, D. S., Deep generative models of genetic variation capture mutation effects (2017), arXiv preprint arXiv:.06527
[99] Liu, D.; Tan, Y.; Yu, Z., Training deep neural networks for the inverse design of nanophotonic structures, ACS Photonics, 5, 4, 1365-1369 (2017)
[100] Yao, K.; Unni, R.; Zheng, Y., Intelligent nanophotonics: merging photonics and artificial intelligence at the nanoscale (2018), arXiv preprint arXiv:1810.11709
[101] Tahersima, M. H.; Kojima, K.; Koike-Akino, T.; Jha, D.; Wang, B.; Lin, C.; Parsons, K., Deep neural network inverse design of integrated nanophotonic devices (2018), arXiv preprint arXiv:1809.03555
[102] Fukushima, K.; Miyake, S., Neocognitron: A self-organizing neural network model for a mechanism of visual pattern recognition, (Competition and Cooperation in Neural Nets (1982), Springer), 267-285
[103] LeCun, Y.; Bengio, Y., Convolutional networks for images, speech, and time series, Handb. Brain Theory Neural Netw., 3361, 10, 1995 (1995)
[104] Lippmann, R., An introduction to computing with neural nets, IEEE Assp Mag., 4, 2, 4-22 (1987)
[105] Yegnanarayana, B., Artificial Neural Networks (2009), PHI Learning Pvt. Ltd.
[106] Nielsen, M. A., Neural Networks and Deep Learning (2015), Determination Press
[107] Hassoun, M. H., Fundamentals of Artificial Neural Networks (1995), MIT press · Zbl 0850.68271
[108] Ballard, D. H., Modular learning in neural networks, (AAAI (1987)), 279-284
[109] Gallinari, P.; LeCun, Y.; Thiria, S.; Fogelman-Soulie, F., Memoires associatives distribuees, Proc. Cogn., 87, 93 (1987)
[110] Le Cun, Y.; Fogelman-Soulié, F. J.I., Modèles Connexionnistes de L’Apprentissage, 2 (1), 114-143 (1987)
[111] Rumelhart, D. E.; Hinton, G. E.; Williams, R. J., Learning Internal Representations By Error Propagation (1985), California Univ San Diego La Jolla Inst for Cognitive Science
[112] Vincent, P.; Larochelle, H.; Bengio, Y.; Manzagol, P.-A., Extracting and composing robust features with denoising autoencoders, (Proceedings of the 25th International Conference on Machine Learning (2008), ACM), 1096-1103
[113] Bengio, Y.; LeCun, Y., Scaling learning algorithms towards AI, Large-scale Kernel Mach., 34, 5, 1-41 (2007)
[114] Bengio, Y., Learning deep architectures for AI, Found. Trends Mach. Learn., 2, 1, 1-127 (2009) · Zbl 1192.68503
[115] L. Theis, W. Shi, A. Cunningham, F. Huszár, Lossy image compression with compressive autoencoders, arXiv preprint arXiv:.00395 (2017).
[116] J. Ngiam, A. Khosla, M. Kim, J. Nam, H. Lee, A.Y. Ng, Multimodal deep learning, in: Proceedings of the 28th international conference on machine learning (ICML-11), 2011, pp. 689-696.
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.