Thomas Mejer Hansen, Christopher C Finlay

The solution to a probabilistic inverse problem is the posterior probability distribution for which a full analytic expression is rarely possible. Sampling methods are therefore often used to generate a sample from the posterior. Decision-makers may be interested in the probability of features related to model parameters (for example existence of a pollution or the cumulative clay thickness) rather than the individual realizations themselves. Such features and their associated uncertainty, are simple to compute once a sample from the posterior distribution has been generated. However, sampling methods are often associated with high computational costs, especially when the prior and posterior distribution is non-trivial (non-Gaussian), and when the inverse problem is non-linear. Here we demonstrate how to use a neural network to directly estimate posterior statistics of continuous or discrete features of the posterior distribution. The method is illustrated on a probabilistic inversion of airborne EM data from Morrill Nebraska, where the forward problem is nonlinear and the prior information is non-Gaussian. Once trained the application of the network is fast, with results similar to those obtained using much slower sampling methods.