  Understanding systems by forward and inverse modeling is a recurrent topic of
research in many domains of science and engineering. In this context, Monte
Carlo methods have been widely used as powerful tools for numerical inference
and optimization. They require the choice of a suitable proposal density that
is crucial for their performance. For this reason, several adaptive importance
sampling (AIS) schemes have been proposed in the literature. We here present an
AIS framework called Regression-based Adaptive Deep Importance Sampling
(RADIS). In RADIS, the key idea is the adaptive construction via regression of
a non-parametric proposal density (i.e., an emulator), which mimics the
posterior distribution and hence minimizes the mismatch between proposal and
target densities. RADIS is based on a deep architecture of two (or more) nested
IS schemes, in order to draw samples from the constructed emulator. The
algorithm is highly efficient since employs the posterior approximation as
proposal density, which can be improved adding more support points. As a
consequence, RADIS asymptotically converges to an exact sampler under mild
conditions. Additionally, the emulator produced by RADIS can be in turn used as
a cheap surrogate model for further studies. We introduce two specific RADIS
implementations that use Gaussian Processes (GPs) and Nearest Neighbors (NN)
for constructing the emulator. Several numerical experiments and comparisons
show the benefits of the proposed schemes. A real-world application in remote
sensing model inversion and emulation confirms the validity of the approach.
