The paper introduces an original permutation approach to assessment of the predictive ability of models based on artificial neural networks. Three models based on artificial neural networks (a multilayer perceptron, a network with the radial basis function, and a neural network with the generalized regression) were implemented to illustrate the permutation approach. Data on the spatial distribution of copper in the upper soil layer of Novy Urengoy (Yamalo-Nenets Autonomous Okrug, Russia) were used for modeling. To evaluate the performance of the models, three different methods were used: error indices estimation, a graphical approach (the Taylor diagram), and a randomization estimation of the probability of obtaining the divergence between the observed and predicted series under assumption that both these datasets are taken from the same population. In the permutation approach, two statistics were used: the difference in means and the correlation coefficient. The permutation approach proved to be productive, as it allowed assessing the significance of the divergence between the observed and predicted datasets.