Content deleted Content added
this is a book, not a journal paper |
No edit summary |
||
Line 39:
=== Kolmogorov network ===
The [[Kolmogorov–Arnold representation theorem]] is similar in spirit. Indeed, certain neural network families can directly apply the Kolmogorov–Arnold theorem to yield a universal approximation theorem. [[Robert Hecht-Nielsen]] showed that a three-layer neural network can approximate any continuous multivariate function.<ref>{{Cite journal |last=Hecht-Nielsen |first=Robert |date=1987 |title=Kolmogorov's mapping neural network existence theorem |url=https://cir.nii.ac.jp/crid/1572543025788928512 |journal=Proceedings of International Conference on Neural Networks, 1987 |volume=3 |pages=11–13}}</ref> This was extended to the discontinuous case by Vugar Ismailov.<ref>{{cite journal |last1=Ismailov |first1=Vugar E. |date=July 2023 |title=A three layer neural network can represent any multivariate function |journal=Journal of Mathematical Analysis and Applications |volume=523 |issue=1 |pages=127096 |arxiv=2012.03016 |doi=10.1016/j.jmaa.2023.127096 |s2cid=265100963}}</ref> In 2024, Ziming Liu and co-authors showed a practical application.<ref>{{cite arXiv |last1=Liu |first1=Ziming |title=KAN: Kolmogorov-Arnold Networks |date=2024-05-24 |eprint=2404.19756 |last2=Wang |first2=Yixuan |last3=Vaidya |first3=Sachin |last4=Ruehle |first4=Fabian |last5=Halverson |first5=James |last6=Soljačić |first6=Marin |last7=Hou |first7=Thomas Y. |last8=Tegmark |first8=Max|class=cs.LG }}</ref>
=== Reservoir computing and quantum reservoir computing===
In reservoir computing a sparse recurrent neural network with fixed weights equipped of fading memory and echo state property is followed by a trainable output layer. Its universality has been demonstrated separately for what concerns networks of rate neurons <ref>{{Cite journal |last=Maass |first=Wolfgang |last2=Markram |first2=Henry |date=2004 |title=On the computational power of circuits of spiking neurons |url=http://www.igi.tugraz.at/maass/psfiles/135.pdf |journal=Journal of computer and system sciences |volume=69 |pages=593–616}}</ref> and spiking neurons, respectively. <ref>{{Cite journal |last1=Grigoryeva |first1=L. |last2=Ortega |first2=J.-P. |date=2018 |title=Echo state networks are universal |journal=Neural Networks |volume=108 |issue=1 |pages=495–508 |arxiv=1806.00797 |doi=10.1016/j.neunet.2018.08.025}}</ref> In 2024, the framework has been generalized and extended to quantum reservoirs where the reservoir is based on qubits defined over Hilbert spaces. <ref>{{cite arXiv |last1=Monzani |first1=Francesco |title=Universality conditions of unified classical and quantum reservoir computing |date=2024|eprint=2401.15067 |last2=Prati |first2=Enrico |class=quant-ph }}</ref>
=== Variants ===
| |||