As　an　important　part　of　machine　learning,　deep　learning　has　been　intensively　used　in　various　fields　relevant　to　data　science.　Despite　of　its　popularity　in　practice,　it　is　still　of　challenging　to　compute　the　optimal　parameters　of　a　deep　neural　network,　which　has　been　shown　to　be　NP-hard.　We　devote　the　present　paper　to　an　analysis　of　deep　neural　networks　with　nonatomic　congestion　games,　and　expect　that　this　can　inspire　the　computation　of　optimal　parameters　of　deep　neural　networks.　We　consider　a　deep　neural　network　with　linear　activation　functions　of　the　form　x+　b　for　some　biases　b　that　need　not　be　zero.　We　show　under　mild　conditions　that　learning　the　weights　and　the　biases　is　equivalent　to　computing　the　social　optimum　flow　of　a　nonatomic　congestion　game.　When　the　deep　neural　network　is　for　classification,　then　the　learning　is　even　equivalent　to　computing　the　equilibrium　flow.　These　results　generalize　a　recent　seminar　work　by　[18],　who　have　shown　similar　results　for　deep　neural　networks　of　linear　activation　functions　with　zero　biases.　©　2021,　Springer　Nature　Switzerland　AG.