Deep-learning based reconstruction of the shower maximum Xmax using the water-Cherenkov detectors of the Pierre Auger Observatory

The successful installation, commissioning, and operation of the Pierre Auger Observatory would not have been possible without the strong commitment and effort from the technical and administrative staff in Malargue. We are very grateful to the following agencies and organizations for financial support. Argentina - Comision Nacional de Energia Atomica; Agencia Nacional de Promocion Cientifica y Tecnologica (ANPCyT); Consejo Nacional de Investigaciones Cientificas y Tecnicas (CONICET); Gobierno de la Provincia de Mendoza; Municipalidad de Malargue; NDM Holdings and Valle Las Lenas; in gratitude for their continuing cooperation over land access; Australia -the Australian Research Council; Brazil - Conselho Nacional de Desenvolvimento Cientifico e Tecnologico (CNPq); Financiadora de Estudos e Projetos (FINEP); Fundacao de Amparo a Pesquisa do Estado de Rio de Janeiro (FAPERJ); Sao Paulo Research Foundation (FAPESP) Grants No. 2019/10151-2, No. 2010/07359-6 and No. 1999/05404-3; Ministerio da Ciencia, Tecnologia, Inovacoes e Comunicacoes (MCTIC); Czech Republic Grant No. MSMT CR LTT18004, LM2015038, LM2018102, CZ.02.1.01/0.0/0.0/16_013/0001402, CZ.02.1.01/0.0/0.0/18_046/0016010 and CZ.02.1.01/0.0/0.0/17_049/0008422; France -Centre de Calcul IN2P3/CNRS; Centre National de la Recherche Scientifique (CNRS); Conseil Regional Ile-de-France; Departement Physique Nucleaire et Corpusculaire (PNC-IN2P3/CNRS); Departement Sciences de l'Univers (SDU-INSU/CNRS); Institut Lagrange de Paris (ILP) Grant No. LABEX ANR-10-LABX-63 within the Investissements d'Avenir Programme Grant No. ANR-11-IDEX-0004-02; Germany-Bundesministerium fur Bildung und Forschung (BMBF); Deutsche Forschungsgemeinschaft (DFG); Finanzministerium Baden-Wurttemberg; Helmholtz Alliance for Astroparticle Physics (HAP); Helmholtz-Gemeinschaft Deutscher Forschungszentren (HGF); Ministerium fur Innovation, Wissenschaft und Forschung des Landes Nordrhein-Westfalen; Ministerium fur Wissenschaft, Forschung und Kunst des Landes Baden-Wurttemberg; Italy - Istituto Nazionale di Fisica Nucleare (INFN); Istituto Nazionale di Astrofisica (INAF); Ministero dell'Istruzione, dell'Universita e della Ricerca (MIUR); CETEMPS Center of Excellence; Ministero degli Affari Esteri (MAE); Mexico-Consejo Nacional de Ciencia y Tecnologia (CONACYT) No. 167733; Universidad Nacional Autonoma de Mexico (UNAM); PAPIIT DGAPA-UNAM; The Netherlands - Ministry of Education, Culture and Science; Netherlands Organisation for Scientific Research (NWO); Dutch national e-infrastructure with the support of SURF Cooperative; Poland - Ministry of Science and Higher Education, grant No. DIR/WK/2018/11; National Science Centre, Grants No. 2013/08/M/ST9/00322, No. 2016/23/B/ST9/01635 and No. HARMONIA 5-2013/10/M/ST9/00062, UMO-2016/22/M/ST9/00198; Portugal - Portuguese national funds and FEDER funds within Programa Operacional Factores de Competitividade through Fundacao para a Ciencia e a Tecnologia (COMPETE); Romania-Romanian Ministry of Education and Research, the Program Nucleu within MCI (PN19150201/16N/2019 and PN-19060102) and project PNIII-P1-1.2-PCCDI-2017-0839/19PCCDI/2018 within PNCDI III; Slovenia - Slovenian Research Agency, grants P1-0031, P1-0385, I0-0033, N1-0111; Spain - Ministerio de Economia, Industria y Competitividad (FPA2017-85114-P and PID2019-104676GB-C32, Xunta de Galicia (ED431C 2017/07), Junta de Andalucia (SOMM17/6104/UGR, P18-FR-4314) Feder Funds, RENATA Red Nacional Tematica de Astroparticulas (FPA2015-68783-REDT) and Maria de Maeztu Unit of Excellence (MDM-2016-0692); USA -Department of Energy, Contracts No. DE-AC02-07CH11359, No. DE-FR02-04ER41300, No. DE-FG02-99ER41107 and No. DE-SC0011689; National Science Foundation, Grant No. 0450696; The Grainger Foundation; Marie Curie-IRSES/EPLANET; European Particle Physics Latin American Network; and UNESCO.The atmospheric depth of the air shower maximum X-max is an observable commonly used for the determination of the nuclear mass composition of ultra-high energy cosmic rays. Direct measurements of X-max are performed using observations of the longitudinal shower development with fluorescence telescopes. At the same time, several methods have been proposed for an indirect estimation of X-max from the characteristics of the shower particles registered with surface detector arrays. In this paper, we present a deep neural network (DNN) for the estimation of X-max The reconstruction relies on the signals induced by shower particles in the ground based water-Cherenkov detectors of the Pierre Auger Observatory. The network architecture features recurrent long short-term memory layers to process the temporal structure of signals and hexagonal convolutions to exploit the symmetry of the surface detector array. We evaluate the performance of the network using air showers simulated with three different hadronic interaction models. Thereafter, we account for long-term detector effects and calibrate the reconstructed X-max using fluorescence measurements. Finally, we show that the event-by-event resolution in the reconstruction of the shower maximum improves with increasing shower energy and reaches less than 25 g/cm(2) at energies above 2x10(19) eV.United States Department of Energy (DOE) DE-AC02-07CH11359 - DE-FR02-04ER41300 - DE- FG02-99ER41107 - DE-SC0011689Argentina - Comision Nacional de Energia AtomicaArgentina - Agencia Nacional de Promocion Cientifica y Tecnologica (ANPCyT)Argentina - Consejo Nacional de Investigaciones Cientificas y Tecnicas (CONICET)Argentina - Gobierno de la Provincia de MendozaArgentina - Municipalidad de MalargueAustralian Research CouncilConselho Nacional de Desenvolvimento Cientifico e Tecnologico (CNPQ) Fundacao de Apoio a Pesquisa do Distrito Federal (FAPDF)Brazil - Financiadora de Estudos e Projetos (FINEP)Fundacao Carlos Chagas Filho de Amparo a Pesquisa do Estado do Rio De Janeiro (FAPERJ)Fundacao de Amparo a Pesquisa do Estado de Sao Paulo (FAPESP) 2019/10151-2-2010/07359-6-1999/05404-3Brazil - Ministerio da Ciencia, Tecnologia, Inovacoes e Comunicacoes (MCTIC)Czech Republic Grant MSMT CR LTT18004 - LM2015038 - LM2018102 - CZ.02.1.01/0.0/0.0/16_013/0001402 - CZ.02.1.01/0.0/0.0/18_046/0016010 - CZ.02.1.01/0.0/0.0/17_049/0008422France - Centre de Calcul IN2P3/CNRSCentre National de la Recherche Scientifique (CNRS)France - Conseil Regional Ile-de-FranceFrance - Departement Physique Nucleaire et Corpusculaire (PNC-IN2P3/CNRS)France - Departement Sciences de l'Univers (SDU-INSU/CNRS)French National Research Agency (ANR) LABEX ANR-10-LABX-63 - ANR-11-IDEX-0004-02Federal Ministry of Education & Research (BMBF)German Research Foundation (DFG)Germany-Finanzministerium Baden-WurttembergGermany-Helmholtz Alliance for Astroparticle Physics (HAP)Germany-Helmholtz-Gemeinschaft Deutscher Forschungszentren (HGF)Germany-Ministerium fur Innovation, Wissenschaft und Forschung des Landes Nordrhein-WestfalenGermany-Ministerium fur Wissenschaft, Forschung und Kunst des Landes Baden-WurttembergItaly - Istituto Nazionale di Fisica Nucleare (INFN)Italy - Istituto Nazionale di Astrofisica (INAF)Italy - Ministero dell'Istruzione, dell'Universita e della Ricerca (MIUR)Italy - CETEMPS Center of ExcellenceItaly - Ministero degli Affari Esteri (MAE)Consejo Nacional de Ciencia y Tecnologia (CONACyT) 167733Mexico-Universidad Nacional Autonoma de Mexico (UNAM)Mexico-PAPIIT DGAPA-UNAMThe Netherlands - Ministry of Education, Culture and ScienceNetherlands Organization for Scientific Research (NWO)The Netherlands - SURF CooperativePoland - Ministry of Science and Higher Education DIR/WK/2018/11Poland - National Science Centre 2013/08/M/ST9/00322 - 2016/23/B/ST9/01635 -HARMONIA 5-2013/10/M/ST9/00062 - UMO-2016/22/M/ST9/00198Portugal - Portuguese national fundsPortugal - FEDER funds within Programa Operacional Factores de Competitividade through Fundacao para a Ciencia e a Tecnologia (COMPETE)Romania-Romanian Ministry of Education and ResearchRomania-Program Nucleu within MCI PN19150201/16N/2019 - PN-19060102Romania-PNCDI III PNIII-P1-1.2-PCCDI-2017-0839/19PCCDI/2018Slovenian Research Agency - Slovenia P1-0031 - P1-0385 - I0-0033 - N1-0111Spain - Ministerio de Economia, Industria y Competitividad FPA2017-85114-P - PID2019-104676GB-C32Spain - Xunta de Galicia ED431C 2017/07Junta de Andalucia SOMM17/6104/UGR - P18-FR-4314Spain - RENATA Red Nacional Tematica de Astroparticulas FPA2015-68783-REDTSpain - Maria de Maeztu Unit of Excellence MDM-2016-0692National Science Foundation (NSF) 0450696USA - Grainger FoundationUSA - Marie Curie-IRSES/EPLANETUSA - European Particle Physics Latin American NetworkUSA - UNESCOArgentina - NDM Holdings and Valle Las LenasThe Netherlands - Dutch national e-infrastructureSpain - Feder Fund