In this study, we address the inverse scattering problem in the field of acoustics and explore the reconstruction method of cavity shape under acoustically soft boundary conditions. By incorporating an adaptive neuro-fuzzy inference system (ANFIS), an inversion technique based on a single point source and a finite number of measurement points is developed for reconstructing the geometry of acoustically soft cavities. First, the measured near-field data are downscaled by principal component analysis (PCA) method to reduce the complexity of the near-field data. Then, this paper proposes an adaptive fuzzy inference system, which uses gradient descent method and extended Kalman filter (EKF) algorithm to update the pre and post parameters of the model, so as to minimize the mean square error of the retrieved shape parameters. We demonstrate the effectiveness of the method with several numerical examples, verifying that choosing a suitable measurement profile and aperture range will lead to better results of the inversion. Finally, we compare the effectiveness of ANFIS with that of a conventional feed-forward neural network (FNN) for cavity inversion, and verify that ANFIS is robust to noise in scattered field measurements.