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This paper demonstrates the ability of the neural network trained on frequency-sweeping signals with different amplitudes to reconstruct the flame nonlinear response. The neural network architecture consists of a decreasing sequence increasing dimension (DSID) model and a sequence model; the latter one uses the Long short-term memory (LSTM) and encoder of Transformer, respectively. Results show that the neural network trained using the proposed sweeping method with limited training data can reconstruct realistic signals over the envisaged range of frequencies and amplitudes. The nonlinear flame responses obtained by the neural network are further embedded into the closed-loop thermoacoustic feedback to quantify the reconstruction performance of sequence signals. It is demonstrated that the neural network can accurately capture the evolution of the limit cycle. This paper has also compared the effect of different types and sizes of datasets on trained neural networks model; results show that the models trained with our proposed datasets perform better. For small-size datasets, LSTM performs significantly better than encoder of Transformer. Encoder of Transformer is more suitable for large-size datasets.

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Cylindrical ducts with axial mean temperature gradient and mean flows are typical elements in rocket engines, can combustors, and afterburners. Accurate analytical solutions for the acoustic waves of the longitudinal and transverse modes within these ducts can significantly improve the performance of low order acoustic network models for analyses of acoustic behaviours and combustion instabilities in these kinds of ducts. Here, we derive an acoustic wave equation as a function of pressure perturbation based on the linearised Euler equations (LEEs), and the modified WKB approximation method is applied to derive analytical solutions based on very few assumptions. The eigenvalue system is built based on the proposed solutions and applied to predict the resonant frequencies and growth rate for transverse modes. Validations of the proposed solutions are performed by comparing them to the numerical results directly calculated from the LEEs. Good agreements are found between analytical reconstruction and numerical results of three-dimensional transverse modes. The system with both mean temperature profile and mean flow presents a larger absolute value of the growth rate than the condition of either uniform mean temperature or no mean flow.