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©ESO 2020. Artículo firmado por 29 autores. We thank an anonymous referee for helpful comments that improved the quality of this paper. CARMENES is an instrument for the Centro Astronómico Hispano-Alemán de Calar Alto (CAHA, Almería, Spain). CARMENES is funded by the German Max-Planck-Gesellschaft (MPG), the Spanish Consejo Superior de Investigaciones Científicas (CSIC), European Regional Development Fund (ERDF) through projects FICTS-2011- 02, ICTS-2017-07-CAHA-4, and CAHA16-CE-3978, and the members of the CARMENES Consortium (Max-Planck-Institut für Astronomie, Instituto de Astrofísica de Andalucía, Landessternwarte Königstuhl, Institut de Ciències de l’Espai, Insitut für Astrophysik Göttingen, Universidad Complutense de Madrid, Thüringer Landessternwarte Tautenburg, Instituto de Astrofísica de Canarias, Hamburger Sternwarte, Centro de Astrobiología and Centro Astronómico Hispano-Alemán), with additional contributions by the Spanish Ministry of Economy, the German Science Foundation through the Major Research Instrumentation Programme and DFG Research Unit FOR2544 “Blue Planets around Red Stars”, the Klaus Tschira Stiftung, the states of BadenWürttemberg and Niedersachsen, and by the Junta de Andalucía. We acknowledge financial support from NASA through grant NNX17AG24G, the Agencia Estatal de Investigación of the Ministerio de Ciencia through fellowship FPU15/01476, Innovación y Universidades and the ERDF through projects PID2019-109522GB-C51/2/3/4, AYA2016-79425-C3-1/2/3-P and AYA2018- 84089, the Fundação para a Ciência e a Tecnologia through and ERDF through grants UID/FIS/04434/2019, UIDB/04434/2020 and UIDP/04434/2020, PTDC/FIS-AST/28953/2017, and COMPETE2020 - Programa Operacional Competitividade e Internacionalização POCI-01-0145-FEDER-028953., Existing and upcoming instrumentation is collecting large amounts of astrophysical data, which require efficient and fast analysis techniques. We present a deep neural network architecture to analyze high-resolution stellar spectra and predict stellar parameters such as effective temperature, surface gravity, metallicity, and rotational velocity. With this study, we firstly demonstrate the capability of deep neural networks to precisely recover stellar parameters from a synthetic training set. Secondly, we analyze the application of this method to observed spectra and the impact of the synthetic gap (i.e., the difference between observed and synthetic spectra) on the estimation of stellar parameters, their errors, and their precision. Our convolutional network is trained on synthetic PHOENIX-ACES spectra in different optical and near-infrared wavelength regions. For each of the four stellar parameters, Teff, log g, [M/H], and v sin i, we constructed a neural network model to estimate each parameter independently. We then applied this method to 50 M dwarfs with high-resolution spectra taken with CARMENES (Calar Alto high-Resolution search for M dwarfs with Exo-earths with Near-infrared and optical Échelle Spectrographs), which operates in the visible (520–960 nm) and near-infrared wavelength range (960–1710 nm) simultaneously. Our results are compared with literature values for these stars. They show mostly good agreement within the errors, but also exhibit large deviations in some cases, especially for [M/H], pointing out the importance of a better understanding of the synthetic gap., Ministerio de Ciencia e Innovación (MICINN)/FEDER, Consejo Superior de Investigaciones Científicas (CSIC)/FEDER, Fundação para a Ciência e a Tecnologia/FEDER, NASA, COMPETE2020 - Programa Operacional Competitividade e Internacionalização, the German Max-Planck-Gesellschaft (MPG), the Max-Planck-Institut für Astronomie, Instituto de Astrofísica de Andalucía, Landessternwarte Königstuhl, Institut de Ciències de l’Espai, Insitut für Astrophysik Göttingen, Universidad Complutense de Madrid, Thüringer Landessternwarte Tautenburg, Instituto de Astrofísica de Canarias, the German Science Foundation through the Major Research Instrumentation Programme, Hamburger Sternwarte, Centro de Astrobiología and Centro Astronómico Hispano-Alemán, the Spanish Ministry of Economy, the DFG Research Unit FOR2544 “Blue Planets around Red Stars”, the Klaus Tschira Stiftung, the states of BadenWürttemberg and Niedersachsen, the Junta de Andalucía, Spain, the Agencia Estatal de Investigación of the Ministerio de Ciencia, Depto. de Física de la Tierra y Astrofísica, Fac. de Ciencias Físicas, TRUE, pub