Here, Jeff Dean (Google's chief scientist) explains here how the idea behind TPUs was born at Google.
In the early 2010s, he noticed that new speech recognition and vision were getting much better, but they were extremely heavy to run on regular CPUs.
His back-of-the-envelope calculation was: if speech recognition got good enough that, say, 100M people talked to their phone for 3 minutes a day, and all of that ran on CPU servers, Google might need roughly 2x the total number of machines it already had, just for that one feature.
That was obviously insane in cost, power, and time to deploy, so he concluded they needed a completely different kind of hardware, purpose built for neural nets.
Also given neural nets mostly use a small set of simple numeric operations and are very tolerant of low precision and noise, so specialized chips could be dramatically more efficient than CPUs or GPUs.
Instead of using flexible but inefficient FPGAs, he pushed for a custom ASIC: TPUv1, an inference chip on a PCIe card, then convinced the CFO to deploy about $50M worth of them in data centers before every use case was even fully mapped out.
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From 'Radical Ventures' YT channel (link in comment)