NVIDIA is certainly ruling the roost because of CUDA+deep learning, but advantages in chips don't tend to last, and I do not see why this time would be any different. Could be a very interesting few years in the space.
Intel's real competitors are TSMC and GlobalFoundries, who are architecture agnostic. TSMC's market cap is ~90% of Intel's; GF are private, but they're probably in the same ballpark.
Intel is in a unique (and uniquely challenging) position as both an IP and fab company. This vertical integration has proved highly profitable for Intel, but it is also highly risky. Their recent decision to start manufacturing ARM chips points at this uncertain future - diversification is a high priority if they are to reliably recoup the costs of <10nm fabs.
The desktop market that Intel dominate is dying a slow death. Intel has failed miserably in the mobile market and are under threat on several fronts in the server and HPC market. x86 has been a tremendous asset for Intel, but it could easily become a hindrance. Intel is really two very different businesses (IP and fab) that are shackled together.
Intel is great but their current HPC/X86 offerings aren't that great. Xeon Phi and the Xeon SOC's might eventually come out swinging, but currently they aren't.
AMD on the GPU side is pretty stagnant it has almost no answer to the Tesla based servers, and on the software side NVIDIA's software stack which includes CUDA is simply better. On the CPU side we need to see where Zen leads, but even then AMD will have a competitor to the traditional Xeon platform rather than HPC.
ARM is probably more scalable especially since you can easily create semi-custom SOC's for HPC and put a lot of pretty powerful and considerably smaller cores in a server. Intel may get to that point with the Xeon products but it is not there yet.
If you go ask people working on car aerodynamics or geophysics or reservoir modelling for the oil industry, they're mostly all CPU cluster users still.
In the more "smaller" enterprise level HPC market NVIDIA is kinda ruling today, there is almost no point of building a traditional CPU based HPC platform if you are an enterprise (or even a small/medium size education/research institution).
Intel is trying to get into that business with it's Xeon SOC's and Xeon Phi compute hardware but I don't know how much traction they are getting, NVIDIA probably by now pretty much controls all of that market and Intel and AMD are more or less rounding errors.
ARM can really give NVIDIA a fight if they can provide semi-custom ASIC's for HPC you can easily shove 100's of cores into a single server making it potentially comparable to NVIDIA's HPC offerings.