Model selection by maximizing the marginal likelihoods of candidate physics-based models of a thermoacoustic experiment.

We automate an experiment consisting of an electrical heater inside a vertical tube. We examine 16 heater positions, 8 powers, and various cold configurations. For each configuration we force acoustically and measure the response with 8 probe microphones. We introduce prior knowledge that the acoustic response depends on how the heat release rate depends on the velocity at the heater and propose several physics-based models. We find the most likely values of each model's parameters, given the 17600 datapoints, using first order adjoints. We obtain the uncertainties in each model's parameters using Laplace's method using second order adjoints. We identify the most plausible model by calculating the integral of the posterior probability over parameter space (marginal likelihood). This penalizes models with too many parameters, which over-fit the data. We find the maximum marginal likelihood, allowing the measurement noise to vary, in order to account for model error elsewhere. This method successfully ranks the candidate models. The most plausible model is a simple n-tau model, with two parameters, which fits the data well across the entire range without overfitting. The physical insight from this method exposes systematic experimental error that otherwise might not be noticed.