With the increasing uncertainties from weather-dependent distributed generation and the user behavior-driven electrified loads within distribution networks, more frequent real-time analysis of network operating states will be required. Feasible operation region (FOR), which characterizes the range of operating states of a distribution network within which no network constraints are violated, provides an effective tool for such analysis. However, the existing physical-law-based model(PLM) for FORs suffers from inaccuracies due to simplifying assumptions used to derive explicit relationships between nodal power injections and node voltages or line currents. This paper develops a more precise analytical FOR model using a data-driven approach that leverages accurate simulation results. Two regression models are proposed: one directly constructs the FOR model from simulation-derived datasets for the FOR boundaries, while the other quantifies the PLM error and compensates for it using a linearized error model trained on the same dataset. The effectiveness of the proposed models is validated using the 11kV benchmark distribution networks from the UK Generic Distribution System (UKGDS). Compared with the PLM, both regression models exhibit substantial improvements in accuracy. In assessing the operational feasibility of the network, the proposed models achieve an accuracy exceeding 99% and eliminate 69% of misjudged operating states when using the PLM. Furthermore, the regression models are highly efficient, requiring approximately three orders of magnitude less computation time than conventional power flow simulations and linear power flow models, respectively. Although the proposed data-driven approach relies on computationally intensive offline simulations to generate datasets for the real FOR boundaries, this limitation can be addressed through more efficient simulation techniques or parallel computing.