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Machine learning is now used in many areas of astrophysics, from detecting exoplanets in Kepler transit signals to removing telescope systematics. Recent work demonstrated the potential of using machine learning algorithms for atmospheric retrieval by implementing a random forest to perform retrievals in seconds that are consistent with the traditional, computationally-expensive nested-sampling retrieval method. We expand upon their approach by presenting a new machine learning model, \texttt{plan-net}, based on an ensemble of Bayesian neural networks that yields more accurate inferences than the random forest for the same data set of synthetic transmission spectra. We demonstrate that an ensemble provides greater accuracy and more robust uncertainties than a single model. In addition to being the first to use Bayesian neural networks for atmospheric retrieval, we also introduce a new loss function for Bayesian neural networks that learns correlations between the model outputs. Importantly, we show that designing machine learning models to explicitly incorporate domain-specific knowledge both improves performance and provides additional insight by inferring the covariance of the retrieved atmospheric parameters. We apply \texttt{plan-net} to the Hubble Space Telescope Wide Field Camera 3 transmission spectrum for WASP-12b and retrieve an isothermal temperature and water abundance consistent with the literature. We highlight that our method is flexible and can be expanded to higher-resolution spectra and a larger number of atmospheric parameters.
