They aren't really directly comparable like you imply. A memristor is basically a resistor that changes resistance when you run current through it. An FPGA is a clever layout of many thousands of logic gates that can be programmed to form arbitrarily complex digital logic circuits. It's like comparing apples to skyscrapers because they're both associated with New York.

Presumably, a memristor based neural network would have the advantage over an FPGA of requiring significantly less silicon area to achieve the same function. I imagine an FPGA based neural network would approximate analog signals digitally, perhaps using floating point "half's" or something. Memristors would directly operate on analog signals, encoding information as amplitudes or pulses of currents and voltages.

Notably, FPGAs and GPUs can't really be directly compared to each other in terms of power consumption unless you specify specific use cases. You can't build the equivalent of a mainstream GPU out of FPGAs without severely limiting the clock speed (because of how physically large it would be), and if you did anyway, it would use many orders of magnitude more power to function. So, a GPU is way more power efficient than an FPGA for rendering graphics. There are certainly problems that a GPU isn't good at solving, and so there's a good chance that an FPGA solution would be more power efficient.

Correct me if I'm wrong, but isn't it a good metaphor to say that FPGAs are just ASIC emulators? I.e. much less efficient than an equivalent ASIC, but good for prototyping and something to use in production if you can't afford to manufacture your own chip (which is most low-volume use cases). Under this metaphor, memristor-based ASICs will obviously be more efficient than FPGAs emulating them, but that's predicated on there being a high-volume use case justifying the creation of a memristor chip in the first place.

Wow there... sometimes digital emulation of an analog circuit is faster than the analog circuit since you can solve for equilibria, easily repeat calculations, use well researched programming tools, etc

My understanding of the use of FPGAs and ASICs that are used to speed up neural networks (such as those in phones) is that they are simply designed to do the types of calculations used for NNs more quickly (matrix operations) and generally at a reduced level of precision. This is very different from a memristor approach where the structure of the network itself would be represented in the silicon. I also think it's unfair to compare the two because it took decades of work to get CMOS transistors to where they are today. I imagine that once commercial applications for memristors appear many optimizations/improvements will present themselves.

"the structure of the network itself would be represented in the silicon" -- ASICs then? perhaps even hybrid analog/digital one, where fixed coefficients are stored in digital memory, while input data is analog.

I believe there is a great value in being able to "snapshot" the state and later load exactly the same state into millions of devices. And I cannot see how this will easily work with memristors.