Nonlinear inverse problems usually have no analytical solution and may be solved by Monte Carlo methods that create a set of samples, representative of the a posteriori distribution. We show how neural networks can be trained on these samples to give a continuous approximation to the inverse relation in a compact and computationally efficient form. We examine the strengths and weaknesses of this approach and use it to determine the full a posteriori distribution of crustal thickness from surface wave velocities. The solution to this inverse problem shows significant asymmetry and large uncertainties due to trade-off with shear velocity structure around the Moho. We produce maps of maximum likelihood crustal thickness across Eurasia which are in agreement with current knowledge about the crust; thus we provide an independent confirmation of these models. In this application, characterized by repeated inversion of similar data, the neural network algorithm proves to be very efficient. ¿ 1999 American Geophysical Union