Geophysical inversion is challenging because realistic models of the earth are nonlinear and their corresponding inverse problems are ill-posed due to non-existent or nonunique solutions that are sensitive to small changes in the data. There is now a significant body of work in geophysics that uses neural networks to represent an inverse operator, however, it is difficult to obtain sufficient observational data sets to train a neural network because field data acquisition is both time consuming and costly. Therefore it has become common to create training data with forward models that capture physical laws, empirically validated rules or other domain expertise. In this work we extend traditional approaches to Bayesian inference to the problem of generating training data from a geophysical forward model. Regularization techniques are used to inform prior information in Bayesian inference which forms the training samples. The result is an ensemble of parameter estimates that are propagated through a nonlinear forward model and used as labeled training data in supervised learning. The trained neural network is used to map a set of observations to a set of geophysical parameters.