The development of mathematical models, and the associated numerical simulations, are challenging in higher-dimensional systems featuring flows of supercritical fluids in various applications. In this paper, a data-driven methodology is presented to achieve system order reduction, and to identify important physical information within the principal flow features. Firstly, a new hybrid neural network based on radial basis function (RBF) and multi-layer perceptron (MLP) methods, namely RBF-MLP, is tested to achieve a highly nonlinear approximation. When provided with 1000 nonlinear test samples, this model provides an excellent prediction accuracy with a maximum regression coefficient (R) of 0.99 and a minimum root mean square error (RMSE) below 1%. Furthermore, the model is also proven to be flexible enough to capture accurately the turbulent fluctuation characteristics, even at significant nonlinear buoyancy conditions. Secondly, the high-dimensional buoyancy data is collected and integrated into a matrix database. Subsequently, a proper orthogonal decomposition (POD) approach is employed to reduce the high-dimensional database, and to obtain a set of low-dimensional POD basis-spanned space, which defines a reduced-order system with low-dimensional basis vectors. The results reveal that the first five order modes contain dominant flow features, accounting for 93% of the total mode energy, which can be selected to reconstruct the physical flow field. Thirdly, a new data-driven POD-ANN model is established to construct the nonlinear mapping between the full-field buoyancy data and decomposed basis vectors. It is also demonstrated that the POD-ANN model reconstructs the principal flow features accurately and reliably. This POD-ANN model can be used to provide new insights for reduced-order modelling and for reconstructing physical fields of higher-dimensional nonlinear flow cases.