Thermoacoustic instabilities, driven by the interaction between the heat release rate from the combustion process with the combustion chamber pressure waves, have long been a problem in jet and rocket engine design. Unacceptably large oscillations often appear during full-scale engine tests, despite being absent during part-scale tests, leading to costly re-designs. A physics-informed, data-driven model of a flame would allow for important quantities, such as the fluctuating heat release rate of the combustion process, to be estimated for a given burner geometry. This in turn would enable different geometries to be assessed for susceptibility to thermoacoustic instabilities before any physical testing, ensuring a cheaper design process.

In this thesis, data from artificial flame simulations and laboratory experiments are assimilated into increasingly complex physics-based models of the flame front. These physics-based models, based on the G-equation, are qualitatively correct. and their physical parameters must be inferred from the data to render them quantitatively accurate. For the artificial and laboratory Bunsen flames, the ensemble Kalman filter (EnKF) infers the parameters of the physics-based model and their uncertainties from a sequence of images. The method is reliable but computationally expensive: it takes hours for the EnKF to converge to parameter estimates and uncertainties for each test case. An alternative method is proposed, which uses a heteroscedastic Bayesian neural network ensemble (BayNNE) trained on a library of simulated flame fronts with known parameters to infer the parameters and uncertainties of the physics-based model. Generating the library of simulated flame fronts and training the BayNNE on the library are both computationally expensive. However, once trained, the BayNNE infers the parameters and uncertainties from an input sequence of six flame front images in milliseconds, which is fast enough to be used in real-time applications. The BayNNE method is applied to data from a version of the Volvo burner. The Volvo flame is noisy, exhibits transient behaviour and is observed over a limited spatial window, which means that the EnKF is unable to converge. On the other hand, the BayNNE is able to robustly infer the parameters and uncertainties from a sequence of ten flame front images. Using the physics-based model with the inferred parameters, the flame front is extrapolated downstream of the observation window. This allows for the n and τ fields of a distributed n-τ model for the heat release of the burner to be calculated. These fields are entered into a Helmholtz solver to predict the growth rates and frequencies of the thermoacoustic system. The BayNNE method is then applied to the same data but with a more physically-informed model of the velocity field using the discrete vortex method. Although this thesis's conclusions for thermoacoustic behaviour are unsurprising, the BayNNE method is a potentially cheap way to combine sparse experimental measurements with complete numerical results and can readily be extended to other experiments and to other models.

As increasingly large quantities of experimental data become available, the researcher's challenge is to extract useful information without becoming overwhelmed by the quantity of data. Assimilation into physics-based models, as performed in this thesis, is attractive because the models are physically-interpretable and extrapolatable. The work presented shows one possible approach, and how it can be applied to a particularly long-standing problem in engineering: the modelling, and ultimately the control of, thermoacoustic instabilities.