Parameter Estimation for Spatio-Temporal Models using Bayesian Optimisation and Gaussian Processes*

With many physical models having only numerical solutions, it can be challenging to fit predictions to data. We show how accurate joint estimation of parameters can be achieved efficiently with Bayesian optimization and Gaussian processes, even for spatio-temporal models, using the Cahn-Hilliard equation of phase separation as an exemplar.
To find globally optimal parameters, we represent the distance between the results of simulation and some observed outcome using a loss function. Instead of a computationally expensive grid-based search for the minimum loss, we adopt a Bayesian optimisation approach placing a Gaussian process over the loss, representing the function as an infinite-dimensional normal distribution that can be used to estimate it over its entire input space, with quantified uncertainty. The GP can be trained using a small number of evaluations, and a suggestion for the next test-point can be obtained automatically. New evaluations are used to update the GP estimation. Bayesian optimisation has the ability to localise regions where minima occur within a small number of iterations. The intrinsic incorporation of uncertainty allows for an effective trade-off between this exploitation and exploration of the wider input space.