The geometric shape of turbine blades varies due to manufacturing scatter and wear. The influence of these variations on the stresses can be quantified using probabilistic methods. This requires several finite element analyses. However, to obtain statistically reliable results, a large number of deterministic calculations are required. In this context, the use of deep neural networks (DNN) is a promising approach to replace finite element calculations. Unlike traditional surrogate models such as polynomials, DNNs are able to use data from multiple nodes simultaneously. They can exploit the interdependence of result values from locally nearby nodes without the need for a topologically identical mesh in the finite element analysis (FEA). This is a significant advantage over conventional surrogate models that must be evaluated at topologically identical positions.To introduce and validate the approach, a probabilistic analysis of an FEA model of a beam is performed. Training and benchmark data are generated by means of a Monte Carlo simulation (MCS). The separate benchmark data is used to validate the model. A sensitivity analysis using the Coefficient of Importance serves as a demonstration example of probabilistic methods. The developed method is then applied to local results on an evaluation mesh of a cooled turbine blade model. A sensitivity study is carried out with the validated model and the results are compared with a previous publication.Finally, a DNN model is trained on the MCS data of the turbine blade model without using an evaluation mesh. Good agreement is obtained with the approximation of a new parameter set using this model. It is shown that the DNN is a promising surrogate model for probabilistic analysis.