Thermoacoustic instabilities can be highly detrimental to the operation of aircraft gas turbine combustors within design conditions, and hence their prediction and suppression are crucial. This work uses a Bayesian machine learning method to infer the parameters of a bluff-body stabilised, physics-informed flame model in real-time. The flame front is modelled using the $G$-equation, a level-set method which segments the flow into regions of reactants and products. The flow past the bluff-body is modelled with a discrete vortex method (DVM) to account for vortical perturbations on the flame front. Using the physics-informed model with the learned parameters from both the $G$-equation and the DVM, a flame transfer function (FTF) is obtained, from which the growth rates of instability in the system can be calculated. A heteroscedastic Bayesian neural network ensemble (BayNNE) is trained on a library of flame front simulations with known target parameters in both models. The trained BayNNE is a surrogate model for a Bayesian posterior of the target parameters given the input flame front coordinates. The ensemble predicts some parameters of the DVM with more certainty than others, showing which are more influential in affecting the flame front. Using the learned posterior, the flame fronts are re-simulated, to extrapolate the flame beyond the experimental window where it was observed. Flame results are also extrapolated in parameter space. These extrapolated flame shapes are then used to calculate thermoacoustic frequencies and growth rates of the system. We observe that the growth rates and frequencies do not show a strong dependency on the amplitude of forcing, which is one of the inferred parameters of the physics-informed model. This important result suggests that a FTF derived at high amplitude, when it is observable, is also valid at low amplitude, when it is not observable.