{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,6,30]],"date-time":"2026-06-30T23:48:17Z","timestamp":1782863297102,"version":"3.54.5"},"reference-count":50,"publisher":"Elsevier BV","license":[{"start":{"date-parts":[[2026,12,1]],"date-time":"2026-12-01T00:00:00Z","timestamp":1796083200000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/www.elsevier.com\/tdm\/userlicense\/1.0\/"},{"start":{"date-parts":[[2026,12,1]],"date-time":"2026-12-01T00:00:00Z","timestamp":1796083200000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/www.elsevier.com\/legal\/tdmrep-license"},{"start":{"date-parts":[[2027,6,25]],"date-time":"2027-06-25T00:00:00Z","timestamp":1813881600000},"content-version":"am","delay-in-days":206,"URL":"http:\/\/www.elsevier.com\/open-access\/userlicense\/1.0\/"},{"start":{"date-parts":[[2026,12,1]],"date-time":"2026-12-01T00:00:00Z","timestamp":1796083200000},"content-version":"stm-asf","delay-in-days":0,"URL":"https:\/\/doi.org\/10.15223\/policy-017"},{"start":{"date-parts":[[2026,12,1]],"date-time":"2026-12-01T00:00:00Z","timestamp":1796083200000},"content-version":"stm-asf","delay-in-days":0,"URL":"https:\/\/doi.org\/10.15223\/policy-037"},{"start":{"date-parts":[[2026,12,1]],"date-time":"2026-12-01T00:00:00Z","timestamp":1796083200000},"content-version":"stm-asf","delay-in-days":0,"URL":"https:\/\/doi.org\/10.15223\/policy-012"},{"start":{"date-parts":[[2026,12,1]],"date-time":"2026-12-01T00:00:00Z","timestamp":1796083200000},"content-version":"stm-asf","delay-in-days":0,"URL":"https:\/\/doi.org\/10.15223\/policy-029"},{"start":{"date-parts":[[2026,12,1]],"date-time":"2026-12-01T00:00:00Z","timestamp":1796083200000},"content-version":"stm-asf","delay-in-days":0,"URL":"https:\/\/doi.org\/10.15223\/policy-004"}],"funder":[{"DOI":"10.13039\/501100004377","name":"The Hong Kong Polytechnic University","doi-asserted-by":"publisher","award":["P0053092"],"award-info":[{"award-number":["P0053092"]}],"id":[{"id":"10.13039\/501100004377","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/100000001","name":"National Science Foundation","doi-asserted-by":"publisher","award":["DMS-2309551"],"award-info":[{"award-number":["DMS-2309551"]}],"id":[{"id":"10.13039\/100000001","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/100000001","name":"National Science Foundation","doi-asserted-by":"publisher","award":["DMS-2012860"],"award-info":[{"award-number":["DMS-2012860"]}],"id":[{"id":"10.13039\/100000001","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/100000001","name":"National Science Foundation","doi-asserted-by":"publisher","award":["DMS-2307465"],"award-info":[{"award-number":["DMS-2307465"]}],"id":[{"id":"10.13039\/100000001","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/100000001","name":"National Science Foundation","doi-asserted-by":"publisher","award":["DMS-2510829"],"award-info":[{"award-number":["DMS-2510829"]}],"id":[{"id":"10.13039\/100000001","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/100000001","name":"National Science Foundation","doi-asserted-by":"publisher","award":["DMS-2309530"],"award-info":[{"award-number":["DMS-2309530"]}],"id":[{"id":"10.13039\/100000001","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":["elsevier.com","sciencedirect.com"],"crossmark-restriction":true},"short-container-title":["Neural Networks"],"published-print":{"date-parts":[[2026,12]]},"DOI":"10.1016\/j.neunet.2026.109268","type":"journal-article","created":{"date-parts":[[2026,6,20]],"date-time":"2026-06-20T15:08:39Z","timestamp":1781968119000},"page":"109268","update-policy":"https:\/\/doi.org\/10.1016\/elsevier_cm_policy","source":"Crossref","is-referenced-by-count":0,"special_numbering":"C","title":["Fourier multi-component and multi-layer neural networks: Unlocking high-frequency potential"],"prefix":"10.1016","volume":"204","author":[{"ORCID":"https:\/\/orcid.org\/0000-0003-4115-7891","authenticated-orcid":false,"given":"Shijun","family":"Zhang","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0009-0006-0373-7181","authenticated-orcid":false,"given":"Hongkai","family":"Zhao","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-1537-7364","authenticated-orcid":false,"given":"Yimin","family":"Zhong","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-7647-2600","authenticated-orcid":false,"given":"Haomin","family":"Zhou","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"78","reference":[{"issue":"1","key":"10.1016\/j.neunet.2026.109268_bib0001","doi-asserted-by":"crossref","first-page":"8","DOI":"10.1137\/18M118709X","article-title":"Optimal approximation with sparsely connected deep neural networks","volume":"1","author":"B\u00f6lcskei","year":"2019","journal-title":"SIAM Journal on Mathematics of Data Science"},{"issue":"5","key":"10.1016\/j.neunet.2026.109268_bib0002","doi-asserted-by":"crossref","first-page":"A3285","DOI":"10.1137\/19M1310050","article-title":"A phase shift deep neural network for high frequency approximation and wave problems","volume":"42","author":"Cai","year":"2020","journal-title":"SIAM Journal on Scientific Computing"},{"key":"10.1016\/j.neunet.2026.109268_bib0003","unstructured":"Cai, W., & Xu, Z.-Q. J. (2019). Multi-scale deep neural networks for solving high dimensional PDEs. 10.48550\/arXiv.1910.11710."},{"key":"10.1016\/j.neunet.2026.109268_bib0004","doi-asserted-by":"crossref","first-page":"14","DOI":"10.3389\/fams.2018.00014","article-title":"Construction of neural networks for realization of localized deep learning","volume":"4","author":"Chui","year":"2018","journal-title":"Frontiers in Applied Mathematics and Statistics"},{"key":"10.1016\/j.neunet.2026.109268_bib0005","doi-asserted-by":"crossref","first-page":"303","DOI":"10.1007\/BF02551274","article-title":"Approximation by superpositions of a sigmoidal function","volume":"2","author":"Cybenko","year":"1989","journal-title":"Mathematics of Control, Signals, and Systems"},{"key":"10.1016\/j.neunet.2026.109268_bib0006","series-title":"The thirty-eighth annual conference on neural information processing systems","article-title":"Addressing spectral bias of deep neural networks by multi-grade deep learning","author":"Fang","year":"2024"},{"key":"10.1016\/j.neunet.2026.109268_bib0007","series-title":"International conference on learning representations (ICLR)","article-title":"Multiplicative filter networks","author":"Fathony","year":"2021"},{"key":"10.1016\/j.neunet.2026.109268_bib0008","doi-asserted-by":"crossref","first-page":"259","DOI":"10.1007\/s00365-021-09543-4","article-title":"Approximation spaces of deep neural networks","volume":"55","author":"Gribonval","year":"2022","journal-title":"Constructive Approximation"},{"issue":"05","key":"10.1016\/j.neunet.2026.109268_bib0009","doi-asserted-by":"crossref","first-page":"803","DOI":"10.1142\/S0219530519410021","article-title":"Error bounds for approximations with deep ReLU neural networks in Ws,p norms","volume":"18","author":"G\u00fchring","year":"2020","journal-title":"Analysis and Applications"},{"key":"10.1016\/j.neunet.2026.109268_bib0010","doi-asserted-by":"crossref","DOI":"10.1016\/j.chaos.2024.115620","article-title":"Nonlinear chaotic Lorenz\u2013L\u00fc\u2013Chen fractional order dynamics: A novel machine learning expedition with deep autoregressive exogenous neural networks","volume":"189","author":"Hassan","year":"2024","journal-title":"Chaos, Solitons & Fractals"},{"key":"10.1016\/j.neunet.2026.109268_bib0011","series-title":"2016 IEEE Conference on computer vision and pattern recognition (CVPR)","first-page":"770","article-title":"Deep residual learning for image recognition","author":"He","year":"2016"},{"key":"10.1016\/j.neunet.2026.109268_bib0012","unstructured":"Hendrycks, D., & Gimpel, K. (2016). Gaussian error linear units (GELUs). 10.48550\/arXiv.1606.08415."},{"issue":"2","key":"10.1016\/j.neunet.2026.109268_bib0013","doi-asserted-by":"crossref","first-page":"251","DOI":"10.1016\/0893-6080(91)90009-T","article-title":"Approximation capabilities of multilayer feedforward networks","volume":"4","author":"Hornik","year":"1991","journal-title":"Neural Networks"},{"issue":"5","key":"10.1016\/j.neunet.2026.109268_bib0014","doi-asserted-by":"crossref","first-page":"359","DOI":"10.1016\/0893-6080(89)90020-8","article-title":"Multilayer feedforward networks are universal approximators","volume":"2","author":"Hornik","year":"1989","journal-title":"Neural Networks"},{"key":"10.1016\/j.neunet.2026.109268_bib0015","series-title":"Proceedings of the 32nd international conference on machine learning","first-page":"448","article-title":"Batch normalization: Accelerating deep network training by reducing internal covariate shift","volume":"vol. 37","author":"Ioffe","year":"2015"},{"issue":"4","key":"10.1016\/j.neunet.2026.109268_bib0016","doi-asserted-by":"crossref","first-page":"3635","DOI":"10.1137\/21M144431X","article-title":"Deep neural networks with ReLU-Sine-Exponential activations break curse of dimensionality in approximation on H\u00f6lder class","volume":"55","author":"Jiao","year":"2023","journal-title":"SIAM Journal on Mathematical Analysis"},{"key":"10.1016\/j.neunet.2026.109268_bib0017","series-title":"3rd international conference on learning representations, ICLR 2015, san diego, ca, usa, may 7\u20139, 2015, conference track proceedings","article-title":"Adam: A method for stochastic optimization","author":"Kingma","year":"2015"},{"issue":"5","key":"10.1016\/j.neunet.2026.109268_bib0018","doi-asserted-by":"crossref","first-page":"1970","DOI":"10.4208\/cicp.OA-2020-0179","article-title":"Multi-scale deep neural network (MscaleDNN) for solving Poisson\u2013Boltzmann equation in complex domains","volume":"28","author":"Liu","year":"2020","journal-title":"Communications in Computational Physics"},{"key":"10.1016\/j.neunet.2026.109268_bib0019","series-title":"2024 IEEE\/CVF Conference on computer vision and pattern recognition (CVPR)","first-page":"2713","article-title":"FINER: Flexible spectral-bias tuning in implicit neural representation by variable-periodic activation functions","author":"Liu","year":"2024"},{"issue":"5","key":"10.1016\/j.neunet.2026.109268_bib0020","doi-asserted-by":"crossref","first-page":"5465","DOI":"10.1137\/20M134695X","article-title":"Deep network approximation for smooth functions","volume":"53","author":"Lu","year":"2021","journal-title":"SIAM Journal on Mathematical Analysis"},{"issue":"1","key":"10.1016\/j.neunet.2026.109268_bib0021","doi-asserted-by":"crossref","first-page":"99","DOI":"10.1145\/3503250","article-title":"NeRF: Representing scenes as neural radiance fields for view synthesis","volume":"65","author":"Mildenhall","year":"2021","journal-title":"Communications of the ACM"},{"key":"10.1016\/j.neunet.2026.109268_bib0022","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1016\/j.neunet.2019.12.013","article-title":"Error bounds for deep ReLU networks using the Kolmogorov-Arnold superposition theorem","volume":"129","author":"Montanelli","year":"2020","journal-title":"Neural Networks"},{"key":"10.1016\/j.neunet.2026.109268_bib0023","unstructured":"Morsali, A., Vaez, M., Soltani, H., Kazerouni, A., Taati, B., & Mohammad-Noori, M. (2025). STAF: Sinusoidal trainable activation functions for implicit neural representation. 10.48550\/arXiv.2502.00869."},{"key":"10.1016\/j.neunet.2026.109268_bib0024","series-title":"Proceedings of the 27th international conference on machine learning (ICML-10)","first-page":"807","article-title":"Rectified linear units improve restricted Boltzmann machines","author":"Nair","year":"2010"},{"key":"10.1016\/j.neunet.2026.109268_bib0025","unstructured":"Novello, T., Aldana, D., & Velho, L. (2024). Taming the frequency factory of sinusoidal networks. 10.48550\/arXiv.2407.21121."},{"key":"10.1016\/j.neunet.2026.109268_bib0026","series-title":"Proceedings of the 36th international conference on machine learning","first-page":"5301","article-title":"On the spectral bias of neural networks","volume":"vol. 97","author":"Rahaman","year":"2019"},{"key":"10.1016\/j.neunet.2026.109268_bib0027","doi-asserted-by":"crossref","DOI":"10.1016\/j.bspc.2024.107192","article-title":"Design of intelligent Bayesian regularized deep cascaded NARX neurostructure for predictive analysis of FitzHugh\u2013Nagumo bioelectrical model in neuronal cell membrane","volume":"101","author":"Raja","year":"2025","journal-title":"Biomedical Signal Processing and Control"},{"key":"10.1016\/j.neunet.2026.109268_bib0028","doi-asserted-by":"crossref","DOI":"10.1016\/j.chaos.2025.116149","article-title":"A hybrid neural-computational paradigm for complex firing patterns and excitability transitions in fractional hindmarsh-rose neuronal models","volume":"193","author":"Raja","year":"2025","journal-title":"Chaos, Solitons & Fractals"},{"key":"10.1016\/j.neunet.2026.109268_bib0029","series-title":"2023 IEEE\/CVF Conference on computer vision and pattern recognition (CVPR)","first-page":"18507","article-title":"WIRE: Wavelet implicit neural representations","author":"Saragadam","year":"2023"},{"key":"10.1016\/j.neunet.2026.109268_bib0030","doi-asserted-by":"crossref","first-page":"74","DOI":"10.1016\/j.neunet.2019.07.011","article-title":"Nonlinear approximation via compositions","volume":"119","author":"Shen","year":"2019","journal-title":"Neural Networks"},{"issue":"5","key":"10.1016\/j.neunet.2026.109268_bib0031","doi-asserted-by":"crossref","first-page":"1768","DOI":"10.4208\/cicp.OA-2020-0149","article-title":"Deep network approximation characterized by number of neurons","volume":"28","author":"Shen","year":"2020","journal-title":"Communications in Computational Physics"},{"issue":"4","key":"10.1016\/j.neunet.2026.109268_bib0032","doi-asserted-by":"crossref","first-page":"1005","DOI":"10.1162\/neco_a_01364","article-title":"Deep network with approximation error being reciprocal of width to power of square root of depth","volume":"33","author":"Shen","year":"2021","journal-title":"Neural Computation"},{"issue":"276","key":"10.1016\/j.neunet.2026.109268_bib0033","first-page":"1","article-title":"Deep network approximation: Achieving arbitrary accuracy with fixed number of neurons","volume":"23","author":"Shen","year":"2022","journal-title":"Journal of Machine Learning Research"},{"key":"10.1016\/j.neunet.2026.109268_bib0034","series-title":"Proceedings of the 39th international conference on machine learning","first-page":"19909","article-title":"Deep network approximation in terms of intrinsic parameters","volume":"vol. 162","author":"Shen","year":"2022"},{"key":"10.1016\/j.neunet.2026.109268_bib0035","series-title":"Advances in neural information processing systems","first-page":"7462","article-title":"Implicit neural representations with periodic activation functions","volume":"vol. 33","author":"Sitzmann","year":"2020"},{"issue":"56","key":"10.1016\/j.neunet.2026.109268_bib0036","first-page":"1929","article-title":"Dropout: A simple way to prevent neural networks from overfitting","volume":"15","author":"Srivastava","year":"2014","journal-title":"Journal of Machine Learning Research"},{"key":"10.1016\/j.neunet.2026.109268_bib0037","series-title":"Advances in neural information processing systems","first-page":"7537","article-title":"Fourier features let networks learn high frequency functions in low dimensional domains","volume":"vol. 33","author":"Tancik","year":"2020"},{"key":"10.1016\/j.neunet.2026.109268_bib0038","doi-asserted-by":"crossref","DOI":"10.1016\/j.cma.2021.113938","article-title":"On the eigenvector bias of Fourier feature networks: From regression to solving multi-scale PDEs with physics-informed neural networks","volume":"384","author":"Wang","year":"2021","journal-title":"Computer Methods in Applied Mechanics and Engineering"},{"issue":"5","key":"10.1016\/j.neunet.2026.109268_bib0039","doi-asserted-by":"crossref","first-page":"1746","DOI":"10.4208\/cicp.OA-2020-0085","article-title":"Frequency principle: Fourier analysis sheds light on deep neural networks","volume":"28","author":"Xu","year":"2020","journal-title":"Communications in Computational Physics"},{"key":"10.1016\/j.neunet.2026.109268_bib0040","doi-asserted-by":"crossref","first-page":"103","DOI":"10.1016\/j.neunet.2017.07.002","article-title":"Error bounds for approximations with deep ReLU networks","volume":"94","author":"Yarotsky","year":"2017","journal-title":"Neural Networks"},{"key":"10.1016\/j.neunet.2026.109268_bib0041","series-title":"Proceedings of the 31st conference on learning theory","first-page":"639","article-title":"Optimal approximation of continuous functions by very deep ReLU networks","volume":"vol. 75","author":"Yarotsky","year":"2018"},{"key":"10.1016\/j.neunet.2026.109268_bib0042","series-title":"Proceedings of the 38th international conference on machine learning","first-page":"11932","article-title":"Elementary superexpressive activations","volume":"vol. 139","author":"Yarotsky","year":"2021"},{"key":"10.1016\/j.neunet.2026.109268_bib0043","series-title":"Advances in neural information processing systems","first-page":"13005","article-title":"The phase diagram of approximation rates for deep neural networks","volume":"vol. 33","author":"Yarotsky","year":"2020"},{"key":"10.1016\/j.neunet.2026.109268_bib0044","article-title":"Deep neural network approximation via function compositions","author":"Zhang","year":"2020","journal-title":"PhD Thesis, National University of Singapore"},{"issue":"35","key":"10.1016\/j.neunet.2026.109268_bib0045","first-page":"1","article-title":"Deep network approximation: Beyond ReLU to diverse activation functions","volume":"25","author":"Zhang","year":"2024","journal-title":"Journal of Machine Learning Research"},{"issue":"5","key":"10.1016\/j.neunet.2026.109268_bib0046","doi-asserted-by":"crossref","first-page":"C1059","DOI":"10.1137\/24M1675990","article-title":"Structured and balanced multicomponent and multilayer neural networks","volume":"47","author":"Zhang","year":"2025","journal-title":"SIAM Journal on Scientific Computing"},{"issue":"3","key":"10.1016\/j.neunet.2026.109268_bib0047","doi-asserted-by":"crossref","DOI":"10.1093\/imaiai\/iaaf022","article-title":"Why shallow networks struggle to approximate and learn high frequencies","volume":"14","author":"Zhang","year":"2025","journal-title":"Information and Inference: A Journal of the IMA"},{"issue":"2","key":"10.1016\/j.neunet.2026.109268_bib0048","doi-asserted-by":"crossref","first-page":"787","DOI":"10.1016\/j.acha.2019.06.004","article-title":"Universality of deep convolutional neural networks","volume":"48","author":"Zhou","year":"2020","journal-title":"Applied and Computational Harmonic Analysis"},{"key":"10.1016\/j.neunet.2026.109268_bib0049","unstructured":"Zhu, H., Liu, Z., Zhang, Q., Fu, J., Deng, W., Ma, Z., Guo, Y., & Cao, X. (2024). FINER++: Building a family of variable-periodic functions for activating implicit neural representation. 10.48550\/arXiv.2407.19434."},{"key":"10.1016\/j.neunet.2026.109268_bib0050","series-title":"Advances in neural information processing systems","first-page":"1583","article-title":"Neural networks fail to learn periodic functions and how to fix it","volume":"vol. 33","author":"Ziyin","year":"2020"}],"container-title":["Neural Networks"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/api.elsevier.com\/content\/article\/PII:S0893608026007288?httpAccept=text\/xml","content-type":"text\/xml","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/api.elsevier.com\/content\/article\/PII:S0893608026007288?httpAccept=text\/plain","content-type":"text\/plain","content-version":"vor","intended-application":"text-mining"}],"deposited":{"date-parts":[[2026,6,30]],"date-time":"2026-06-30T22:54:20Z","timestamp":1782860060000},"score":1,"resource":{"primary":{"URL":"https:\/\/linkinghub.elsevier.com\/retrieve\/pii\/S0893608026007288"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2026,12]]},"references-count":50,"alternative-id":["S0893608026007288"],"URL":"https:\/\/doi.org\/10.1016\/j.neunet.2026.109268","relation":{},"ISSN":["0893-6080"],"issn-type":[{"value":"0893-6080","type":"print"}],"subject":[],"published":{"date-parts":[[2026,12]]},"assertion":[{"value":"Elsevier","name":"publisher","label":"This article is maintained by"},{"value":"Fourier multi-component and multi-layer neural networks: Unlocking high-frequency potential","name":"articletitle","label":"Article Title"},{"value":"Neural Networks","name":"journaltitle","label":"Journal Title"},{"value":"https:\/\/doi.org\/10.1016\/j.neunet.2026.109268","name":"articlelink","label":"CrossRef DOI link to publisher maintained version"},{"value":"article","name":"content_type","label":"Content Type"},{"value":"\u00a9 2026 Elsevier Ltd. All rights are reserved, including those for text and data mining, AI training, and similar technologies.","name":"copyright","label":"Copyright"}],"article-number":"109268"}}