I've been out of the field for ten years now, but it's really nice to see improvements to the core physics to this degree.
I'm still skeptical of your supposed two week 500 heights forecast from the ECMWF model. I live near the western Pacific (i.e. the data hole) and it's really easy to find crazy model solutions after 7 days. And I'm pretty sure you weren't looking at the Southern Hemisphere.
You're probably right to be skeptical, for the record I was only a forecaster for a short period of time over ten years ago... didn't even serve my full four year commitment as I volunteered to get out under the Air Force "force shaping" at the time. I was stationed near Rammstein and we created forecasts for Europe. I was referring to the ECMWF ensemble products, specifically.
Where GPUs shine for PDEs is if you have a lot of extra work for each node, for instance if you have complex chemical reactions or thermodynamics, or if you have a high-order method that requires lots of intermediate computations.
If you don't believe me, you can download the PETSc code and test the ViennaCL solvers versus the regular ones.
I also recall the ECMWF had surprisingly accurate long range forecasts based on ensembles. It could predict 500mb heights out two weeks, no sweat.
Re: your comments... My guess is that a gpu isn't suited for use in an operational model due to data access patterns (and possibly not even helpful with the solver). But again, I'm not a computational pde guy. Also, perhaps machine learning would be useful but that would be post-processing or perhaps parameterizing sub-grid phenomenon. There's already a process called model output statistics (MOS) for adjusting raw fields from a weather model.