Fractional-order vortex beams possess Fractional Orbital Angular Momentum (FOAM) modes, which theoretically have the potential to increase transmission capacity infinitely. Therefore, they have significant application prospects in the field of measurement, optical communication and microparticle manipulation. However, when fractional-order vortex beams propagate in free space, the discontinuity of the helical phase makes them susceptible to diffraction in practical applications, thereby affecting the accuracy of OAM mode recognition and severely limiting the use of FOAM-based optical communication. The problem of achieving machine learning recognition of fractional-order vortex beams under diffraction conditions is currently an urgent and unreported issue. This paper proposes a deep learning (DL) method based on ResNet for accurate recognition of the propagation distance and topological charge of fractional-order vortex beam diffraction process. Utilizing both experimental measured and theoretically simulated intensity distributions, a dataset of vortex beam diffraction intensity patterns in atmospheric turbulence environments was created. An improved 101-layer ResNet structure based on transfer learning was employed to achieve accurate and efficient recognition of the FOAM model at different propagation distances. Experimental results show that the proposed method can accurately recognize FOAM modes with a propagation distance of 100 cm, an interval of 5 cm, and a mode spacing of 0.1 under turbulent conditions, with an accuracy of 99.69%. This method considers the effect of atmospheric turbulence during spatial transmission, allowing the recognition scheme to achieve high accuracy even in special environments. It has the distinguishing capability for ultra-fine FOAM modes and propagation distances that traditional methods cannot achieve. This technology can be applied to multidimensional encoding and sensing measurements based on FOAM beam.