Uncertainty errors have been dealt with in the past using probabilistic methods and analytical methods such as the worst case scenario method. More recently analytical bounds for the radius of the solution set have been derived using a posteriori error estimates of the functional type. Probabilistic methods which usually involve Monte-Carlo sampling are computationally very expensive and analytical methods for complex PDE's tend to be either unknown or unsatisfactorily coarse. We suggest using neural networks to approximate errors generated by uncertain data. Specifically, we create neural networks which estimate the radius of the solution set for an uncertain problem. A dataset was made for training such a network in the case of the Poisson equation with an uncertain source term. One training example has as input two source term functions of the Poisson equation and as output the distance between the two different solutions generated by the source terms. Once trained this neural network can be used as a less computationally expensive substitute for a Monte-Carlo sampler. The neural network performs in a comparable way to analytical bounds (which come from functional a posteriori estimates) and a Monte-Carlo sampling computation. This will be attempted for nonlinear PDE's in later research.