There has been some success using Ensemble Kalman Filters (EnKF) to predict hurricanes [1]. I think that these filters sit somewhere between machine learning and deterministic models. The filters are based on a semi-realistic mathematical model but are updated with statistics every few hours so that they can improve their predictions.
As you said, going full on machine learning "sounds like the next step". However, I would disagree that the ideas at the end of the article are unrealistic. If the new technique can beat the Kalman filters it is already useful.
This is a confusing distinction. Aren't lots of machine learning algorithms deterministic? Sure, some include stochastic elements or components, but even those can be deterministic, e.g. by reusing the same seed for the random number generators.
I also don't see them mentioning anything about noise, so what exactly are they trying to achieve? Surely perfect prediction should already be possible, otherwise what are they comparing the output to?
Surely they're discovering hidden (or obvious) regularities based specifically on the process being deterministic.
As a counter example, the Smale Horseshoe [1] is an example of a inherently unpredictable map - it's approximately equivalent to choosing a real number "at random" and attempting to "predict" each digit as you find it. Sure, if some implementation has chosen a number with repeating digits, you can "predict" those digits but this would have nothing to do with being an oracle for algorithm itself.
More broadly, I think it's pretty established that any realistic model of weather is kind of an extension of this concept - it depends unstably on it's initial conditions (small changes in initial conditions result in large changes in final result after some period).
Just start breaking the problem down. We have modestly accurate ideas of air pressure features across most of the world. Not enough to predict every breeze, but enough to predict what days it'll be stormy in a given location about a week in advance, plus or minus a few days, and plus or minus a bit of severity. With continued training, the models can only get more accurate. Personal weather stations are a thing now, and only growing in numbers.
Combine that with radar and satellite data for an even more accurate picture.
Now, if only we could get a log of every (5min? 1min?) reading of every station...
I have my doubts that this can just overcome the fundamental problem of chaos, but it doesn't sound impossible.
A "chaotic system" that is (a) real (b) a system and (c) condition data is a crossable hurdle. Ideas?