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In addition, it has recently been proposed to approximate solutions of high-dimensional partial differential equations (PDEs) by means of stochastic learning problems involving deep ANNs. There are now also a few rigorous mathematical results in the scientific literature which provide error estimates for such deep learning based approximation methods for PDEs. All of these articles provide spatial error estimates for ANN approximations for PDEs but do not provide error estimates for the entire space-time error for the considered ANN approximations. It is the subject of the main result of this article to provide space-time error estimates for deep ANN approximations of Euler approximations of certain perturbed differential equations. Our proof of this result is based (i) on a certain ANN calculus and (ii) on ANN approximation results for products of the form $[0,T]\\times \\mathbb {R}^{d}\\ni (t,x){\\kern -.5pt}\\mapsto {\\kern -.5pt} tx{\\kern -.5pt}\\in {\\kern -.5pt} \\mathbb {R}^{d}$<\/jats:tex-math>\n [<\/mml:mo>\n 0<\/mml:mn>\n ,<\/mml:mo>\n T<\/mml:mi>\n ]<\/mml:mo>\n \u00d7<\/mml:mo>\n \n \n \u211d<\/mml:mi>\n <\/mml:mrow>\n \n d<\/mml:mi>\n <\/mml:mrow>\n <\/mml:msup>\n \u220b<\/mml:mo>\n (<\/mml:mo>\n t<\/mml:mi>\n ,<\/mml:mo>\n x<\/mml:mi>\n )<\/mml:mo>\n \u21a6<\/mml:mo>\n t<\/mml:mi>\n x<\/mml:mi>\n \u2208<\/mml:mo>\n \n \n \u211d<\/mml:mi>\n <\/mml:mrow>\n \n d<\/mml:mi>\n <\/mml:mrow>\n <\/mml:msup>\n <\/mml:math><\/jats:alternatives><\/jats:inline-formula> where $T{\\kern -.5pt}\\in {\\kern -.5pt} (0,\\infty )$<\/jats:tex-math>\n T<\/mml:mi>\n \u2208<\/mml:mo>\n (<\/mml:mo>\n 0<\/mml:mn>\n ,<\/mml:mo>\n \u221e<\/mml:mi>\n )<\/mml:mo>\n <\/mml:math><\/jats:alternatives><\/jats:inline-formula>, $d{\\kern -.5pt}\\in {\\kern -.5pt} \\mathbb {N}$<\/jats:tex-math>\n d<\/mml:mi>\n \u2208<\/mml:mo>\n \u2115<\/mml:mi>\n <\/mml:math><\/jats:alternatives><\/jats:inline-formula>, which we both develop within this article.<\/jats:p>","DOI":"10.1007\/s10444-022-09970-2","type":"journal-article","created":{"date-parts":[[2023,1,11]],"date-time":"2023-01-11T07:02:43Z","timestamp":1673420563000},"update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":16,"title":["Space-time error estimates for deep neural network approximations for differential equations"],"prefix":"10.1007","volume":"49","author":[{"given":"Philipp","family":"Grohs","sequence":"first","affiliation":[]},{"given":"Fabian","family":"Hornung","sequence":"additional","affiliation":[]},{"ORCID":"https:\/\/orcid.org\/0000-0002-9840-3339","authenticated-orcid":false,"given":"Arnulf","family":"Jentzen","sequence":"additional","affiliation":[]},{"ORCID":"https:\/\/orcid.org\/0000-0002-6791-1779","authenticated-orcid":false,"given":"Philipp","family":"Zimmermann","sequence":"additional","affiliation":[]}],"member":"297","published-online":{"date-parts":[[2023,1,11]]},"reference":[{"issue":"5","key":"9970_CR1","doi-asserted-by":"publisher","first-page":"A3135","DOI":"10.1137\/19M1297919","volume":"43","author":"C Beck","year":"2021","unstructured":"Beck, C., Becker, S., Cheridito, P., Jentzen, A., Neufeld, A.: Deep Splitting Method for Parabolic PDEs. 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