To predict the rate and consequences of shrinkage of the earth's mountain glaciers and ice caps, it is necessary to have improved regional-scale models of mountain glaciation and better knowledge of the subglacial topography upon which these models must operate. The problem of estimating glacier ice thickness is addressed by developing an artificial neural network ( ANN) approach that uses calculations performed on a digital elevation model (DEM) and on a mask of the present-day ice cover. Because suitable data from real glaciers are lacking, the ANN is trained by substituting the known topography of ice-denuded regions adjacent to the ice-covered regions of interest, and this known topography is hidden by imagining it to be ice-covered. For this training it is assumed that the topography is flooded to various levels by horizontal lake-like glaciers. The validity of this assumption and the estimation skill of the trained ANN is tested by predicting ice thickness for four 50 km x 50 km regions that are currently ice free but that have been partially glaciated using a numerical ice dynamics model. In this manner, predictions of ice thickness based on the neural network can be compared to the modeled ice thickness and the performance of the neural network can be evaluated and improved. From the results, thus far, it is found that ANN depth estimates can yield plausible subglacial topography with a representative rms elevation error of +/- 70 m and remarkably good estimates of ice volume.