  Bayesian optimal experimental design (OED) seeks experiments that maximize
the expected information gain (EIG) in model parameters. Directly estimating
the EIG using nested Monte Carlo is computationally expensive and requires an
explicit likelihood. Variational OED (vOED), in contrast, estimates a lower
bound of the EIG without likelihood evaluations by approximating the posterior
distributions with variational forms, and then tightens the bound by optimizing
its variational parameters. We introduce the use of normalizing flows (NFs) for
representing variational distributions in vOED; we call this approach vOED-NFs.
Specifically, we adopt NFs with a conditional invertible neural network
architecture built from compositions of coupling layers, and enhanced with a
summary network for data dimension reduction. We present Monte Carlo estimators
to the lower bound along with gradient expressions to enable a gradient-based
simultaneous optimization of the variational parameters and the design
variables. The vOED-NFs algorithm is then validated in two benchmark problems,
and demonstrated on a partial differential equation-governed application of
cathodic electrophoretic deposition and an implicit likelihood case with
stochastic modeling of aphid population. The findings suggest that a
composition of 4--5 coupling layers is able to achieve lower EIG estimation
bias, under a fixed budget of forward model runs, compared to previous
approaches. The resulting NFs produce approximate posteriors that agree well
with the true posteriors, able to capture non-Gaussian and multi-modal features
effectively.
