We present a machine learning framework for accelerated simulation of supercritical fluid flows by training a deep feedforward neural network (DFNN) to estimate the real-fluid thermophysical properties at the projected conditions of interest. This DFNN model, which replaces the compute-intensive, real-fluid state equations, is coupled to a density-based, finite-volume solver with a preconditioned dual-time-stepping technique in which the primitive variables are adopted as the independent variables in the pseudo-time iteration loop. The DFNN is designed to take in, at each time step, the updated temperature and chemical composition and output (while assuming a constant background pressure) all relevant thermophysical properties that are used by the subsequent routines in the solver. The proposed methodology is implemented and tested in a fully-compressible, large-eddy simulation of supercritical turbulent mixing between gaseous oxygen and liquid kerosene in a swirl coaxial rocket injector. The accuracy and efficiency of the neural network are examined in both the a priori and a posteriori settings.