We extract pixel-level masks of extreme weather patterns using variants of Tiramisu and DeepLabv3+ neural networks. We describe improvements to the software frameworks, input pipeline, and the network training algorithms necessary to efficiently scale deep learning on the Piz Daint and Summit systems. The Tiramisu network scales to 5300 P100 GPUs with a sustained throughput of 21.0 PF/s and parallel efficiency of 79.0%. DeepLabv3+ scales up to 27360 V100 GPUs with a sustained throughput of 325.8 PF/s and a parallel efficiency of 90.7% in single precision. By taking advantage of the FP16 Tensor Cores, a half-precision version of the DeepLabv3+ network achieves a peak and sustained throughput of 1.13 EF/s and 999.0 PF/s respectively.