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However, current approaches to efficiently train these models are limited by in silico emulation on digital computers. Here we develop a method called fully forward mode (FFM) learning, which implements the compute-intensive training process on the physical system. The majority of the machine learning operations are thus efficiently conducted in parallel on site, alleviating numerical modelling constraints. In free-space and integrated photonics, we experimentally demonstrate optical systems with state-of-the-art performances for a given network size. FFM learning shows training the deepest optical neural networks with millions of parameters achieves accuracy equivalent to the ideal model. It supports all-optical focusing through scattering media with a resolution of the diffraction limit; it can also image in parallel the objects hidden outside the direct line of sight at over a kilohertz frame rate and can conduct all-optical processing with light intensity as weak as subphoton per pixel (5.40\u2009\u00d7\u200910<jats:sup>18<\/jats:sup>-\u00a0operations-per-second-per-watt energy efficiency) at room temperature. Furthermore, we prove that FFM learning can automatically search non-Hermitian exceptional points without an analytical model. FFM learning not only facilitates orders-of-magnitude-faster learning processes, but can also advance applied and theoretical fields such as deep neural networks, ultrasensitive perception and topological photonics.<\/jats:p>","DOI":"10.1038\/s41586-024-07687-4","type":"journal-article","created":{"date-parts":[[2024,8,7]],"date-time":"2024-08-07T16:19:01Z","timestamp":1723047541000},"page":"280-286","update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":128,"title":["Fully forward mode training for optical neural networks"],"prefix":"10.1038","volume":"632","author":[{"given":"Zhiwei","family":"Xue","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-8850-6839","authenticated-orcid":false,"given":"Tiankuang","family":"Zhou","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Zhihao","family":"Xu","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Shaoliang","family":"Yu","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-7043-3061","authenticated-orcid":false,"given":"Qionghai","family":"Dai","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-3552-0367","authenticated-orcid":false,"given":"Lu","family":"Fang","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"297","published-online":{"date-parts":[[2024,8,7]]},"reference":[{"key":"7687_CR1","doi-asserted-by":"publisher","first-page":"255","DOI":"10.1126\/science.aaa8415","volume":"349","author":"MI Jordan","year":"2015","unstructured":"Jordan, M. 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