That's more or less what we have now. Slide rules and computers are linear systems which can compute nonlinear functions. The problem is that computing anything

That's more or less what we have now. Slide rules and computers are linear systems which can compute nonlinear functions. The problem is that computing anything takes time and must be a linear set of instructions executed in series (multiplied by many parallel cores). Using an analog approach could be vastly more efficient as operations are inherently parallel. You can fire off every neuron in a layer simultaneously and produce a result within nanoseconds, for basically any number of neurons. You could probably do the entire network as a single atomic operation, but that's a bit beyond my knowledge of neural networks