This paper develops a data-driven reduced-order model of the viscous Moore-Greitzer (MG) partial differential equations (PDEs) by threading together ideas from dimensionality reduction to sparse regression and compressed sensing. Numerical simulation of the infinite dimensional viscous MG system is reduced into low dimensional data using principal component analysis (PCA) and autoencoder neural networks based dimensionality reduction methods. Based on the observation that MG equations close to bifurcations have a sparse representation (normal forms) with respect to high-dimensional polynomial spaces, we use the Sparse Identification of Nonlinear Dynamics (SINDy) algorithm which uses a collection of all monomials as a sampling matrix and a sparse regression technique to recover a system of two sparse ordinary differential equations (ODEs) with cubic nonlinearities.