  We introduce a novel strategy for machine-learning-based fast simulators,
which is the first that can be trained in an unsupervised manner using observed
data samples to learn a predictive model of detector response and other
difficult-to-model transformations. Across the physical sciences, a barrier to
interpreting observed data is the lack of knowledge of a detector's imperfect
resolution, which transforms and obscures the unobserved latent data. Modeling
this detector response is an essential step for statistical inference, but
closed-form models are often unavailable or intractable, requiring the use of
computationally expensive, ad-hoc numerical simulations. Using particle physics
detectors as an illustrative example, we describe a novel strategy for a fast,
predictive simulator called Optimal Transport based Unfolding and Simulation
(OTUS), which uses a probabilistic autoencoder to learn this transformation
directly from observed data, rather than from existing simulations. Unusually,
the probabilistic autoencoder's latent space is physically meaningful, such
that the decoder becomes a fast, predictive simulator for a new latent sample,
and has the potential to replace Monte Carlo simulators. We provide
proof-of-principle results for $Z$-boson and top-quark decays, but stress that
our approach can be widely applied to other physical science fields.
