  Low-order thermoacoustic models are qualitatively correct, but they are
typically quantitatively inaccurate. We propose a time-domain bias-aware method
to make qualitatively low--order models quantitatively (more) accurate. First,
we develop a Bayesian ensemble data assimilation method for a low-order model
to self-adapt and self-correct any time that reference data becomes available.
Second, we apply the methodology to infer the thermoacoustic states and heat
release parameters on the fly without storing data (real-time). We perform twin
experiments using synthetic acoustic pressure measurements to analyse the
performance of data assimilation in all nonlinear thermoacoustic regimes, from
limit cycles to chaos, and interpret the results physically. Third, we propose
practical rules for thermoacoustic data assimilation. An increase, reject,
inflate strategy is proposed to deal with the rich nonlinear behaviour; and
physical time scales for assimilation are proposed in non-chaotic regimes (with
the Nyquist-Shannon criterion) and in chaotic regimes (with the Lyapunov time).
Fourth, we perform data assimilation using data from a higher-fidelity model.
We introduce an echo state network to estimate in real-time the forecast bias,
which is the model error of the low-fidelity model. We show that (i) the
correct acoustic pressure, parameters, and model bias can be accurately
inferred; (ii) the learning is robust as it can tackle large uncertainties in
the observations (up to 50% the mean values); (iii) the uncertainty of the
prediction and parameters is naturally part of the output; and (iv) both the
time-accurate solution and statistics can be successfully inferred. Data
assimilation opens up new possibility for real-time prediction of
thermoacoustics by synergistically combining physical knowledge and
experimental data.
