A variant of the U-Net convolutional neural network architecture is proposed to estimate linear elastic compatibility stresses in α-Zr (hcp) polycrystalline grain structures. Training data was generated using VGrain software with a regularity α of 0.73 and uniform random orientation for the grain structures and ABAQUS to evaluate the stress fields using the finite element method. The initial dataset contains 200 samples with 20 held from training for validation. The network gives speedups of around 200x to 6000x using a CPU or GPU, with significant memory savings, compared to finite element analysis with a modest reduction in accuracy of up to 10%. Network performance is not correlated with grain structure regularity or texture, showing generalisation of the network beyond the training set to arbitrary Zr crystal structures. Performance when trained with 200 and 400 samples was measured, finding an improvement in accuracy of approximately 10% when the size of the dataset was doubled.