In　this　work,　we　propose　adaptive　deep　learning　approaches　based　on　normalizing　flows　for　solving　fractional　Fokker–Planck　equations　(FPEs).　The　solution　of　a　FPE　is　a　probability　density　function　(PDF).　Traditional　mesh-based　methods　are　ineffective　because　of　a　unbounded　computation　domain,　a　large　number　of　dimensions　and　a　nonlocal　fractional　operator.　To　this　end,　we　represent　the　solution　with　an　explicit　PDF　model　induced　by　a　flow-based　deep　generative　model,　which　constructs　a　transport　map　from　a　simple　distribution　to　the　target　distribution.　We　consider　two　methods　to　approximate　the　fractional　Laplacian.　One　method　is　the　Monte　Carlo　approximation.　The　other　method　is　to　construct　an　auxiliary　model　with　Gaussian　radial　basis　functions　(GRBFs)　to　approximate　the　solution　such　that　we　may　take　advantage　of　the　fact　that　the　fractional　Laplacian　of　a　Gaussian　is　known　analytically.　Based　on　these　two　different　ways　for　the　approximation　of　the　fractional　Laplacian,　we　propose　two　models　to　approximate　stationary　FPEs　and　one　model　to　approximate　time-dependent　FPEs.　To　further　improve　the　accuracy,　we　refine　the　training　set　and　the　approximate　solution　alternately.　A　variety　of　numerical　examples　is　presented　to　demonstrate　the　effectiveness　of　our　adaptive　deep　density　approaches.　©　2023,　The　Author(s),　under　exclusive　licence　to　Springer　Science+Business　Media,　LLC,　part　of　Springer　Nature.