  The estimation of unknown values of parameters (or hidden variables, control
variables) that characterise a physical system often relies on the comparison
of measured data with synthetic data produced by some numerical simulator of
the system as the parameter values are varied. This process often encounters
two major difficulties: the generation of synthetic data for each considered
set of parameter values can be computationally expensive if the system model is
complicated; and the exploration of the parameter space can be inefficient
and/or incomplete, a typical example being when the exploration becomes trapped
in a local optimum of the objection function that characterises the mismatch
between the measured and synthetic data. A method to address both these issues
is presented, whereby: a surrogate model (or proxy), which emulates the
computationally expensive system simulator, is constructed using deep recurrent
networks (DRN); and a nested sampling (NS) algorithm is employed to perform
efficient and robust exploration of the parameter space. The analysis is
performed in a Bayesian context, in which the samples characterise the full
joint posterior distribution of the parameters, from which parameter estimates
and uncertainties are easily derived. The proposed approach is compared with
conventional methods in some numerical examples, for which the results
demonstrate that one can accelerate the parameter estimation process by at
least an order of magnitude.
