In this paper the mixed convection and entropy generation in a square cavity filled with Al2O3-water nanofluid with the presence of a constant axial magnetic field, is analyzed. The upper and bottom walls are adiabatic. Discretization of the governing equations were achieved through a finite volume method and solved with SIMPLE algorithm. In this research the effects of the Rayleigh number (103- 106), Hartmann number (0 - 100) and also inclination angle (0 - 90°) are investigated. When the cavity is rotated, it is observed that the mean Nusselt number and total entropy generation increase when the Rayleigh number increases in cavity. In square cavity, regardless of the Ha number, by increasing of the inclination angel, the mean Nusselt number and entropy generation rate, increase until inclination angel 30°, then decreases. Also when the magnetic field is rotated, it is observed that the mean Nusselt number decrease when the Hartmann number increases. The mean Nusselt number when the cavity rotates with specific inclination angel is less than state that the cavity rotates with specific magnetic field. For finding optimum condition of heat transfer, Artificial Neural networks (ANN) were used. The results from optimization show that as the Rayleigh number increases, the optimum angel decreases. Whatever the Rayleigh number more increases, the decrement in optimum angel more intenses. Also in low the Rayleigh number, as the Hartmann number increases, the optimum angel decreases firstly then increases. In high Rayleigh number, as the Hartmann number increases, the optimum angel increases too.