Bayesian inference of flame impulse responses

The impulse response of a flame to acoustic velocity perturbations is a key quantity for predicting thermoacoustic stability, but its identification from sparse, noisy observations requires solving an ill-posed inverse convolution problem. Standard system identification methods typically proceed in two stages: they first estimate an impulse response from the data, using hand-tuned choices of regularization, model order, and sampling parameters, and then post-process that estimate to extract physically meaningful quantities such as time delays. This process provides no principled mechanism for incorporating prior physical knowledge during inference. In this paper, we instead reformulate the identification problem within a Bayesian framework by imposing from the outset a physically motivated distributed time-delay model in terms of a given number of Gaussian pulses. The corresponding model parameters are inferred directly from the data, providing both the parameters and the impulse response in one step. The number of pulses best supported by the data is then identified using Bayesian model comparison, which balances data fit against model complexity to identify the simplest model capable of explaining the data. The framework is demonstrated on broadband-forced large eddy simulation data from a turbulent swirl-stabilized burner. Bayesian model comparison selects a three-Gaussian impulse response for this flame, consistent with physical interpretations in previous work. Compared with system identification, the Bayesian approach produces impulse responses with fewer spurious features and enables straightforward enforcement of a known low-frequency gain. Finally, we show that the Bayesian approach is robust to significant reductions in signal length, making it appealing for impulse response identification from costly simulations, where there is an incentive to minimize simulation duration. Novelty and significance In this paper, we introduce a Bayesian framework for identifying flame impulse responses from sparse, noisy input-output data. The key novelty is that a physically motivated distributed time-delay model is imposed during inference, so that the impulse response and its underlying delay parameters are identified simultaneously rather than in two separate steps. This replaces the ad hoc choices required in standard system identification with a principled probabilistic formulation. Physical constraints such as causality, smoothness, characteristic convective timescales, and a known low-frequency gain are encoded directly through the model structure and parameter priors. A further novelty is that model order selection is handled automatically through Bayesian model comparison, rather than by manual inspection. These features make the method more interpretable, more physically consistent, and more robust when only short data records are available, which is especially important for costly simulations of turbulent reacting flows.