Orbital　angular　momentum　(OAM)-carrying　beams　have　received　extensive　attention　due　to　their　high-dimensional　characteristics　in　the　context　of　free-space　optical　communication.　However,　accurate　OAM　mode　recognition　still　suffers　from　reference　misalignment　of　lateral　displacement,　beam　waist　size,　and　initial　phase.　Here　we　propose　a　deep-learning　method　to　exquisitely　recognize　OAM　modes　under　misalignment　by　using　an　alignment-free　fractal　multipoint　interferometer.　Our　experiments　achieve　98.35%　recognizing　accuracy　when　strong　misalignment　is　added　to　hyperfine　OAM　modes　whose　Bures　distance　is　0.01.　The　maximum　lateral　displacement　we　added　with　respect　to　the　perfectly　on-axis　beam　is　about　+/-　0.5　beam　waist　size.　This　work　offers　a　superstable　proposal　for　OAM　mode　recognition　in　the　application　of　free-space　optical　communication　and　allows　an　increase　of　the　communication　capacity.　(C)　2021　Chinese　Laser　Press