Is long term weather prediction mathematically impossible ? Are there similar limits to AI and Autonomous systems?

Factual Questions
21

You’ve got a good list of 3 items there. Thank you. Here’s a related fourth requirement:

    Perform those calculations to infinite precision.

Noticing also that your first requirement for "perfect input data" is not only infinite precision as you rightly say, but infinitesimal scale.

Knowing the e.g. temperature of each separate cubic foot of atmosphere isn’t good enough. Neither is knowing the temperature of each individual cubic centimeter. You really need to get down to the near molecular level of knowing what each molecule or few is doing.

And not just as to temperature. We need temperature plus pressure, plus momentum, plus gas mixture, plus pollutants & particulates, plus electrical charge plus …

A tall order to be sure. An impossibly tall order.

22

@Mijin @LSLGuy and other experts

Please comment if I am understanding this correctly :

  1. The uncertainty in predicting momentum and position of a particle at the same time (governed by Pauli’s Uncertainty Principle) is an inherent property of the universe. The product of the uncertainties is going to approach a limit, and cannot be driven to zero, no matter how good the measuring instruments or computational systems become.

  2. The uncertainty in predicting weather is not an inherent property of the “universe”. The uncertainty can be driven to approach zero as better instruments and computational systems are available.

I think I get your assertion and but I’d like to see a proof if it exists. Sorta of like, I agree there are infinite prime numbers but the proof is so much more elegant.

23

Personal NON-expert opinion:

  1. Quantum uncertainty has zero to do with chaotic uncertainty.

  2. Expecting a “proof” in the math/logical sense of weather unpredictability is a categpry error. Kind of like expecting a “proof” of F=ma from physics. It’s not that realm of problem.

24

@LSLGuy - I know it has zero to do . I am presenting it for contrast. What I am asking is if the uncertainty is an inherent quality of the system or is it because of measurement / computational limitations ? And if it is the latter, then is there a proof for it ?

25

IMO, as an amateur meteorologist, my suspicion is it’s both, and then some.

Meteorologists have a pretty good handle on what factors lead to storm creation, intensification, and path, but I think that even they would agree that they don’t have an absolute understanding of everything that plays a role.

Their computer models for forecasting are far better than they used to be, and are continually improving, as the science improves, as they get more access to more data (from weather satellites, and more observation stations), and as computational power improves, but they are nowhere near perfect yet.

The limitations to being able to accurately predict the weather in a particular location weeks or months in advance have little to do with the math, and more to do with a still-imperfect understanding of all of the inputs that affect how weather develops and changes, as well as not having complete data on current conditions: there are many areas of the Earth’s landmass that have few, if any, monitoring devices installed, not to mention the large swathes of area that are covered by oceans. The data on what’s going on in the atmosphere, above the surface, are also important inputs into the models, and that’s data that’s even more sparse today – a primary source for those data points is still weather balloons.

As others have said, because the information that the meteorologists and their models have is limited and incomplete, there will always be a level of error in them, and those errors get compounded as the models attempt to predict further and further into the future.

26

I’m not 100% sure what chaos theory actually had to do with Jurassic Park. Other than it sounds cool and pseudoscientific when the park ultimately descends into chaos.

I guess maybe that there was no way to predict what would happen by introducing “butterflies” into your environment that are 30 ft high, weigh 10 tons, can outrun a jeep, and eat a goat in one gulp.

Not sure if it is the same thing you are asking, but I think with suitably complex autonomous systems, you do have the possibility of emergent behavior. That is to say the system behaving in ways that the inventors didn’t originally intend.

One theoretical example that comes to mind is the “paperclip problem”. The theory goes that you give an AI the command “maximize paperclip production” and without any constraints it starts purposing all energy production and all the mass on the planet to the manufacture of paperclips.

27

I don’t think so.

I think “climate” is what’s overall expected for the area at given seasons – covering even whether the area does have seasons, and what they are – whereas “weather” is what it’s doing (or did do) in particular locations at specific times.

“Climate” might be “this region historically usually got 22 to 29 inches of rain a year, more of it in May and September but some every month, with the maximum/minimum amount in any given day being less than X in 95% of years; but this appears to be changing in that over the previous 10 years the range was 15 to 40 inches a year, it was heavy in March through May and then there was almost no rain in July and August, and the maximum in a given day has repeatedly been greater than X x 2”; whereas “weather” is “We had 4” of rain on my particular farm on May 4th and more is expected tomorrow (and how am I going to get the spring planting done?)."

28

Exactly. As this NWS web page describes the distinction:

Or, more thoroughly:

29

I was going to do this addendum as an edit, but I see there’s already another post inbetween, so here it is on its own:

And a “long-range weather forecast” that amounts to more than the general “climate” sense tells you whether it’s going to rain on this farm, or at least in this county, on May 4 when it’s not May 4 yet. Right now “long-range” is a week or two and very blurry and uncertain at that distance. (Where I am, it’s often uncertain even on the morning of May 4; in part because it may rain on my farm and not on the one next door.) Such long-range forecasts may well improve – those one-to-two week forecasts used to be nonexistent – but improving it to the point at which I can check on January 11 whether it’s going to be raining here on May 4, let alone to the point at which I can check in January 2023 whether it’s going to be raining here on May 4, 2024 or 2034, carries so many practical difficulties that no, it’s not going to happen. You’d have to know, as has been said, all the characteristics of every atom, and of every chemical and organism as an organization of atoms, and, in addition, how all the interactions among all of them affect all the other interactions among all of them.

30

Guess I won’t be skipping work that day to bike up the Little Miami Trail.

31

Nothing; it was something that Crichton came up with in order to have a mathematician babble on about a trendy topic that had gotten a lot of pop-sci press. I think the point it was trying to illustrate is that while the dinosaurs were engineered to be all females but they inserted frog DNA in gaps of genetic material recovered from amber, which allowed the dinosaurs to switch genders because that is an ability some frog species have, which is a consequence that scientists didn’t consider. Never mind that pretty much all of the genetics and microbiology in the novel doesn’t make a lick of sense; the appropriate mathematical and biological disciplines would be risk analysis, complexity theory, and systems biology, not chaos theory which is concerned with the limit of predicting behavior of dynamical systems subject to perturbative effects.

Stranger

32

IANA expert, but I do have a PhD (Philosopher of the Dope :wink: )

Correct. AIUI, descriptions which focus on the practical aspect of bouncing a photon off something to see where it is, say, are misleading. Our best models of physics suggest that particles do not have very precise positions and momenta at the same time. They are not tiny ping pong balls. They’re ghostly clouds of weirdness (this terminology may be a bit advanced if you don’t have a PhD :smiley: )

Well at a certain hypothetical point, the two domains combine. This is slightly different to what LSLGuy said, but I don’t think there’s actually any dispute, as I’ll explain…

What limits the precision of our models today is how many sensors we have, how accurate they are, how much geographical (and atmospheric) volume they cover, how much time resolution they have etc. (And of course how good our models are, but that’s not relevant to the point I’m making right now)
If we improved the quantity and accuracy of our data tenfold, it might only increase how long we can predict the weather to a given accuracy by another day or so, as we’re getting to ever finer precision requirements. And after that it gets much harder still.

That’s why I think that the practical limitation of predicting the weather might just be 10 days say. You get diminishing returns the farther you go.

But, speaking hypothetically / as a thought experiment, the physical limit of predicting the weather would ultimately be at the point of trying to measure individual molecules’ position and momentum. This is nothing like how we make weather predictions today, and probably never will be. But, in this hypothetical scenario, with tech so incredible that it can measure individual molecules (and not just on earth but all of the sun’s particles too)…yeah you could in principle predict the weather a long way into the future, months, maybe years. But not arbitrarily far, because of the uncertainty principle.

33

The point of the Butterfly Effect isn’t that any single butterfly is relevant; it’s that every butterfly is relevant. In principle, with precise enough data, you could reliably forecast the weather three weeks out… but “precise enough data” would include what every butterfly in the world is doing. And, of course, everything else that has as much influence on the weather as a butterfly, like a human waving their hand. In other words, predicting the weather three weeks out is harder than predicting the fine details of behavior of every human on the planet.

34

Did Pauli hit Heisenberg over the head and steal his Uncertainty Principle?

35

:joy: sorry confused Exclusion for Uncertainty

36

That’s OK, I realized too late that I missed the truly clever phrasing: “Did Pauli exclude Heisenberg from his Uncertainty Principle?”

37

From a provably impossible point of view we can reason about the amount of information and computational effort needed. There quickly comes a point where the size of the computation, in state, effort and speed becomes larger than the system being modelled.

It isn’t useful if the computer has a minimum possible size larger than the observable universe and takes longer to compute a prediction than the time it is forecasting for arrives.

This is part of what chaos theory gave us. It provides guidance on just how insanely accurate our computations need to be, and thus the amount of state the computation has to hold. That also places bounds on speed. Basically the fastest way to get a result is to wait and see what happens.

38

Complex systems are not always unpredictable, and chaotic behaviour usually exists within a range of outputs, not across the entire system.

For example, take a double pendulum and swing it hard. At first, its behaviour will be perfectly predictable, because the energy it has will cause it to swing around in a circle. As the energy fades, it will enter a chaotic zone of completely unpredictable behaviour. Later, as the energy drops enough, the thing will rock back and forth, again in a perfectly predictable manner.

I built a double pendulum a couple of years ago, and experimented with releasing it as close to exactly the same way as I could, by using a stopper at the top to hold it in the same position before release. Nevertheless, whatever tiny differences there were between each attempt resulted in wildly different outcomes - the butterfly effect at work.

A good example (not mine)

As regards climate and weather, there are two sources of potential forecasting error: One is the nature of chaos - non-linear feedbacks and sensitivity to initial conditions. The other is the fact that you are trying to predict the future of a system highly affected by other, unpredictable things. Such as how humans behave, flares from the sun, volcanoes, or any number of other things that can’t be foreseen.

But even then, you can predict things about climate that are outside the zone of chaos. Average summer temp will be higher than now, for example. If the sun increased 20% in output, it would be safe to predict that the climate will get warmer.

Then there are things that are not strictly predictable, but highly likely. Such as the world being warmer 50 years from now. Not certain, but a high probability event.

On the short end, we can predict weather events primarily because of satellites and our ability to see weather patterns. If a giant thunderstorm is moving directly towards you, rain is a good prediction, But even then, storms often suddenly veer for no reason, fronts stall out, etc.

Weather reports more than a week or two out are likely just ‘predicting’ a reversion to the mean, If it’s supposed to be 12 degrees on average at this time of year and it’s currently -10, the long range weather foreact will likely just draw a regression line from tye current temp to the average. So a forecast of warmer weather in two or three weeks may not be based on much at all other than a guess at regression to the mean.

No prediction of the future is perfect. It can’t be, since the future is unknown. Especially true for systems that interact with many other complex systems as the climate does.

39

One complicating factor with predicting weather is that all the plants and animals react to changes in the weather, and their actions cause changes in weather. Not only do the predictions have to correctly model all the actions of the inert stuff like air, water, and heat, but they also have to predict what all the living organisms of the world will do in reaction to changes in weather. For example, when it’s hot and dry people water their lawns more, which causes more humidity over the cities, which causes cloud formation, and that causes the weather to change. When the weather is nice people walk more, which means they don’t drive as much, which means the heat and exhaust from cars isn’t entering the atmosphere, and that causes the weather to change. And even if the organisms aren’t reacting to the weather, there’s still the issue of the organisms changing the landscape to suit their needs, such as clearing a forest for farming. There might be some chance of mathematically predicting the weather on an inert planet like Mars, but it’s probably impossible on a planet with living organisms due to their unpredictable behavior.

40

If the AI Autonomous systems are truly digital, then there shouldn’t be any chaos effects to the extent that given the exact same input you should always get the exact same output. So there is no mathematical limits to the accuracy. However if they are complicated, and there is more than one reasonably valid solution to the problem they are trying to solve, then they could exhibit chaos like behavior in that very slightly different inputs can give very different outputs.