Acoustic weighing is a promising method for non-contact mass measurement of tiny objects as it avoids contamination and contact losses. However, due to the highly nonlinear nature of the acoustic field, some parameters of the mechanism model of acoustic weighing cannot be accurately simulated, thereby reducing the accuracy of acoustic weighing. To improve the accuracy of acoustic weighing, we propose an acoustic weighing method based on oscillating signals and feature enhancement network. Firstly, to drive the object oscillation and collect oscillation data, an acoustic levitation-based data acquisition system is constructed. Then, to break the limitations of the mechanism model, a feature enhancement network named CNN-BiLSTM-SE is proposed, which directly establishes the correlation between oscillating signals and actual mass. Finally, these data are used to train and test the proposed network model, validating the effectiveness of the model. Experimental results show that the method achieves high accuracy in measuring object mass, following the actual measurements with remarkable consistency. In addition, our approach is also suitable for acoustic weighing of small and sensitive objects, opening up new perspective for the study and application of nonlinear acoustic systems.