  Machine learning (ML) has often been applied to space weather (SW) problems
in recent years. SW originates from solar perturbations and is comprised of the
resulting complex variations they cause within the systems between the Sun and
Earth. These systems are tightly coupled and not well understood. This creates
a need for skillful models with knowledge about the confidence of their
predictions. One example of such a dynamical system is the thermosphere, the
neutral region of Earth's upper atmosphere. Our inability to forecast it has
severe repercussions in the context of satellite drag and collision avoidance
operations for objects in low Earth orbit. Even with (assumed) perfect driver
forecasts, our incomplete knowledge of the system results in often inaccurate
neutral mass density predictions. Continuing efforts are being made to improve
model accuracy, but density models rarely provide estimates of uncertainty. In
this work, we propose two techniques to develop nonlinear ML models to predict
thermospheric density while providing calibrated uncertainty estimates: Monte
Carlo (MC) dropout and direct prediction of the probability distribution, both
using the negative logarithm of predictive density (NLPD) loss function. We
show the performance for models trained on local and global datasets. This
shows that NLPD provides similar results for both techniques but the direct
probability method has a much lower computational cost. For the global model
regressed on the SET HASDM density database, we achieve errors of 11% on
independent test data with well-calibrated uncertainty estimates. Using an
in-situ CHAMP density dataset, both techniques provide test error on the order
of 13%. The CHAMP models (on independent data) are within 2% of perfect
calibration for all prediction intervals tested. This model can also be used to
obtain global predictions with uncertainties at a given epoch.
