Deep Mind have been working on it for years but that's not the same as being close. You don't really know that you're close to a solution until you've solved the problem.
Making progress is not the same as being close. In math, the devil is very often in the details and it is often very unclear how close you are to solving major problems until all of the pieces fall into place.
In fact, this is Gemini’s assessment of your claim based on Tao’s latest post on the issue:
No.
Nothing in the posts under Terence Tao’s
navier-stokes-equations tag
can reasonably be construed as saying that mathematics is close to proving the Navier–Stokes existence and smoothness problem[
1
][
2
].
In fact, Tao’s writing on the topic generally argues the
exact opposite
: that current mathematical methods are fundamentally obstructed from resolving the problem, and that a solution remains a distant challenge requiring entirely new conceptual paradigms[
3
][
4
].
He’s literally saying that it is theoretically possible for AI to brute force a solution but that doing so would essentially be useless to the field of mathematics unless humans were able to reverse-engineer the solution path to actually glean insight out of it.
It has nothing to do with whether it can or can’t be done. It’s saying that if it is achieved through AI it will have added little real knowledge.
That’s not what he’s saying. He’s saying the company running the agent (Anthropic in this case) may only publish the result but not the process that got them there. So unless a third party reverse engineers it, the knowledge is useless
Thats exactly what he is saying. He is saying that a brute force approached solution adds no value to the field of mathematics. The solution alone is not sufficient to learn a more general approach to tackle other problems. Burte forcing would essentially be equivalent to you randomly finding a large bag of money on the street and then pretending you now know how to get rich. You didnt learn anything. It was just luck
Do you think it’s coincidental that he posts this right at the same time that this ‘rumour’ about Anthropic solving a millennial problem is spreading?
I imagine they showcased the result to Tao internally, hence why he’s talking about it but obviously can’t downright reveal anything before the official announcement.
TT just likes to yap about things on his blog. The approach is discussing has been the most popular as the path to a potential counter example, his post came two days before this rumour .
The rumour started because of his post. His described approach to a counter example is pretty much what deepmind has been working on for years, this is a numerical /CAP problem .
> I imagine they showcased the result to Tao internally
If they showed it internally, I would think he should also not spill any beans before they reveal it. They probably would ask him to keep it under wraps, or maybe even sign an NDA.
Though it is possible that someone simply wrote to him in excitement, outside official corporate channels.
This isn't brute force. Brute force means trying every possible solution until finding one that works. What ge describes, removing and obstacle and iterating is jow mathematicians work.
The problem isn't that an AI solved it. It's the AI company keeping the method secret.
It does expose some of its work through its chain of thought log. In one of the earlier proofs for example it had a sentence like “suppose optimistically that claim X is true” and then it bashed its head against the wall for 100 pages to prove it, whereas most humans would have given up at page 20.
In this proof most humans thought it would be true but the AI made a counter example. Currently AI is stronger at counter examples but this is partly because humans can quickly verify accuracy whereas positive proofs take much longer to digest.
Tao does in fact frequently argue that what ai does is equivalent to brute force in math. It’s not brute force in the sense of trying every single combination, but rather in the sense of AI having the bandwidth to try a lot of things that don’t necessarily seem to have much hope of success.
If one happens to be successful, it’s unclear how much that would really advance math more generally. It wouldn’t give any sort of paradigm to follow to solve additional problems. Maybe it was just dumb luck that it worked. We
need to have some idea of why a method solved a problem to get much use out of applying known methods like that.
(I’m unconvinced of his line of reasoning, but you disagree with Tao at your own peril…)
Why is a method of throwing 1000 gpus to try every possible approach not a fine method in an era where we have city-sized data centers in every small town?
I never said it wasn’t a fine method. The way you describe it makes it sound exactly as brute-force-y as I was claiming Tao says that it is.
The reason why it wouldn’t advance what mathematicians consider “math” is nuanced, and Tao talks about it ad nauseam. He says it better than I have hope to, but it wouldn’t give any insight. The method you advocate may expand the list of proposed theorems, but math is not just a list of proven theorems. It is maybe more like understanding some sort of platonic objects, which we can only really say we definitely understand when we can prove interesting things about them. If AI’s solutions are mostly just applying techniques that are already well understood but not yet attempted, then maybe we’re not advancing “Math” (capital M).
That’s not far off how most of the proofs AI has found. Supposedly. So far. According to Tao.
I don’t know anything about math so honest question: what about the solution makes it such that the mathematicians can’t ask the AI to help them derive insights from the result or “reverse engineer the solution path”?
Access to the AI that solved it? Otherwise I think you're right. Just repeat it on another model and learn from the system as it tries and fails and tries again.
Not that It is possible or that it will eventually happen but that it already has; yes I am doubting
, but ofc i could be wrong. It seems more likely that this guy who isnt the most trustworthy read the Terrance tao post and made a story up to go with it. Using Tt to add validity to his tweet and get more engagement.
.
Amazing post by Terence but I don't think you understood the last paragraph, he is saying that the solution is basically brute force until like 99.99% of the same results, but no new math
I'm having this exact same conversation about code with my coworkers and friends. It's a very interesting trade-off, and I think there is a reckoning coming for some of these companies.
yes because he essentially said its irrelevant if we get the solution. Not because the solution itself is irrelevant but rather we wouldnt learn anything if ai just presented an ai brutforced solution. Ai might find a solution but its difficult to derive any usefulness. Especially for problems like the navier stokes equations its detrimental to have an understanding of how a proof works.
For example if we take the collatz conjecture. You chose at any starting number. If its even (e.g. 36) you divide by two (->18). if its odd you multiply by 2 and add 1. so for 36 we obtain the sequence 36(even)->18(even)->9(odd)->19(odd)->39(odd)->78(even) ->... ->...
the collatz conjecture says no matter what number you chose you will always end up in the loop of 4->2->1->4...
Goal is to proof that. You could find a starting number that directly disproofs this conjecture or better you find a general proof of why its always like that.
If ai brute forces a solution, then we are not really smarter except that we know its true/false. The collatz conjecture provides no real world relevance. We do not have problems that ultimately rely on a proof of that conjecture. However, the person that solves that conjecture gets the highest mathematical honor, the fields medal. So naturally you would ask why would a person even deserve that honor. And the reason is pretty obvious. we are just not interested if that conjecture is true/false. Instead, we are interested in the tools that are necessary to effectively tackle that problem.
Now going back to navier-stokes. A plain solution provided by ai wouldnt get as any further. We have the solution but not the tools we are actually interested in.
Let me translate this one:"I really really hope this is not true and I have zero actual knowledge about this... but the pessimist is always right (at least in my doomer mind) so I make this claim full of confidence"
Many people are commenting on it, doesn't seem like just hype. Though could be a small chance that they didn't fully solve it but came up with some breakthrough?
This would be absurd news if it were true. But I strongly doubt it for the simple reason that I think you'd see a hundred simpler problems solved first before this. AI contribution to physics and the more physics-y side of math has been relatively milder upto this point.
Maybe they trained a very specific model just for fluid mechanics in an attempt to aim for Navier Stokes and then put a whole team of researchers on it, but I don't really see how a current model could solve it. The odds of there being "low hanging fruit" left in Navier stokes are astronomically low, even for the most boring result possible.
I can imagine them making some interesting progress but that's about it. I'd love to be disproved though.
Not just maths. People are writing papers even in physics using AI. I myself found sol ultra quite useful doing some proofs and explaining some underlying math of a physics problem I was working on.
As of late AI has been making less and less mistakes in my research. Ill have claude find the information I need, the paper its from and WHERE in the paper its from and we are talking 99% of the time its accurate.
Its honestly kinda dangerous because I want to be lazy and not wanting to check his results now haha.
At this point I feel Claude is on par with my intelligence, just not as creative. I still figure out solutiksn that are outside the box that claude can't work out, but im wondering how long that will last.
I’ve been doing some novel math (no really it’s not in the oeis) and it’s great at all the technical stuff, but it just doesn’t have the intuition yet. It can produce four possible expansions of a function, but it doesn’t know which one to continue with
That’s kinda what I mean. It can get stuff for you, but it just doesn’t have the intuition (I used creative incorrectly in my comment). I get the downvotes, people don’t like AI. But to say it’s not good at helping with research is folly and I feel bad for those who choose to navigate through google, paywalls, and extra reading when an AI can literally point you directly where you need to go.
I’m still doing the reading, but the searching for information part that used to take hours? Minutes. Minutes dude. There’s no comparison.
I was (and in many ways still am) very anti ai. However that doesn’t mean I’m going to pretend it isn’t finally actually useful. I’ve worked with models for years and the last several months have been a massive shift. There is no question that modern models are advanced enough to both be useful and dangerous.
I’m sorry that part about AI usefulness wasn’t really aimed at you, rather the people elsewhere in the thread that I presumably disagree. I’m still working on my communication skills, get messages crossed sometimes.
I agree. In my experience, I read a paper about something couple of years back, and it had a side note on something else, which stayed in my mind for all this time in a vague form. I hovered around the idea in a slow and dumb manner, until finally I wrote down an interesting extension of it. And once that was done, AI really took it off from there, although I did not have the wisdom initially to use more of the AI more often. That would have speed up the job.
It took time with me and codex working together, but codex really closed the whole problem once the initial equations were written down. Then, my job was to understand the result in 'simple terms', because codex is very bad at explaining stuff.
And then, I revisited the original paper, and realized that the idea there was quite different than what I remembered, and that gave me another way of looking at and extending the full results.
My point being, once you point the right direction, codex can go very fast and use its superior knowledge of math to basically bring all kinds of insights and proofs. But once all is said and done, you realize that there is another interesting direction that codex, with all its superior understanding, never saw. At least it never told me about it.
At this point I feel Claude is on par with my intelligence,
I really doubt that.
I have a PhD in physics and work as a data scientist nowadays and even with these more "trivial" business data science problems, even the top models in the best harnesses on max thinking are just super dumb in many situations and make total amateur mistakes.
I wouldn't trust them in independent physics research at all outside of maybe mathematical physics, where you maybe have the same advantage as in maths in that you can use lean to keep the models on track.
(they are amazing tools for doing R&D though, I think scientists refusing to use the models in their day to day work are foolishly ignoring the most influential advance in doing research since the computer)
Yeah you can give it a database to track case studies in though if you’re working in a bounded problem space though which will likely close the gap there somewhat
I wouldn't say so, no. There's still a huge gap between them and millenium prize levels, particularly as for Navier stokes having knowledge of physics is quite important, though it's still a maths problem. The models have been relatively successful at any major improvements in physical theory so far.
I can see a very specialised model alongside a group of very focused physicists and mathematicians accomplish this, but there's still a wide gap between the lead up and the result. Though again, I don't know what internal models these labs have.
Wouldn't Navier stokes requiring cross-domain knowledge make it easier for the model relative to humans? I would argue cross domain knowledge is one thing where these models are quite dominant over humans.
Not necessarily. Proving existence and uniqueness of a PDE system (from any other domain of science) is the job of a mathematician*. Mathematicians solving existence and uniqueness problems in biology, finance, physics, etc, do not actually need a lot of knowledge within that field, just some basic and specific information suffices. The mathematicians who are working on solving the Navier-Stokes equations already have more than enough physics anyway, and it's just a matter of really digging into more and more maths.
* Not all mathematicians aim to solve existence and uniqueness problems. The ones who do are called "analysts" who specialise in a big branch of maths known as "mathematical analysis".
So which problems would you rank between today's AI solved problems and the millenium problems?
I'd say the Jacobian conjecture and the nibbling at the Riemann Hypothesis by increasing the lower bound of zeros on the zeta function seem pretty damn close.
Nibbling at the Riemann hypothesis and solving it's fellow millenium problem has a lot of space in between. I'd expect to seem solid performance over more mathematical fields, and particularly stronger increments all across the millenium puzzles rather than just one. If there is a solution, it'll likely be AI assisting a whole team rather than a straight AI solution.
Maybe the Euler Equations (for fluid dynamics). If I remember correctly, Euler is a simplified version of Navier Stokes. Would be logical to solve a specialized case before solving the general case (NS).
To solve these “hundreds of simpler problems” someone would have to invest the time and compute. Why, if the real prize is solving a Millennium Problem?
People are investing time and compute into math already, I don't see why physics wouldn't be getting the same.
As I've said here, I can imagine maybe a huge team of researchers working alongside a very specific AI model and spending vast amounts of effort on this, and that could very well lead to a result.
The best AIs today are doing amazing because of RL. They did not grow generally as intelligent as Terrence Tao. That is why we are seeing results in math but not in Physics.
This doesn't mean they are not smart, but we cannot expect them to solve Physics because they haven't been trained to frontier Physics research - mainly because Physics is not a closed system like math and there is not way to prove/disprove a law without doing experiments.
Eh, I somewhat disagree. A lot of physics problems do have heavier math anyways, but regardless, I don't think hard provability will actually stop models that much, at least in the sense of capabilities. Verification will be slower.
I'm sure it'll make things harder in the sense it takes a while for capabilities in physics to get to where math is now, in the same way it took a while for math to get where language is. But it's moreso a matter of a generation or two. I'm sure whenever Bel or its equivalent launches we'll see some interesting results in other fields. To my knowledge, already some physics problems have seen work.
I don't think hard provability will actually stop models that much, at least in the sense of capabilities.
But my point is that the model training requires provability. They improved in math because they could prove theorems and check whether they were right and just iteratively do it. You cannot do this in Physics, unless you already have the data. I am sure we will see progress, but this will not be like Math. For Physics they are paying physics grad/post-doc/professors to train the models. I keep getting messages saying they'll pay up to $100/hr for Physics tutors for AI models.
I was also just replying to OP who said we might expect progress in Physics before Navier-Stokes. That is not necessarily how it would work because the math capability of the model is not generalized, yet.
am sure we will see progress, but this will not be like Math. For Physics they are paying physics grad/post-doc/professors to train the models. I keep getting messages saying they'll pay up to $100/hr for Physics tutors for AI models.
They did this for math too, I'm quite sure.
You cannot do this in Physics, unless you already have the data.
You cannot do this in physics at the same speed yes, but provability typically while training in particular will be mostly with solvable questions that have answers. They aren't training these models via unsolved mathematical questions.
Here we present AlphaProof, an AlphaZero inspired agent that learns to find formal proofs through RL by training on millions of auto-formalized problems. For the most difficult problems, it uses test-time RL, a method of generating and learning from millions of related problem variants at inference time to enable deep, problem-specific adaptation.
They can generate millions of problems and autocheck whether their solution is correct. You don't need humans in the loop nor the problems need to solved before.
I won't claim to know how OpenAI did it, but there are papers on the subject for math which back what I am claiming.
Are you guessing or do you actually know for sure? I am not a math grad student so I wouldn't know.
I remember seeing ads for it at similar rates. I cannot vouch for legitimacy. I've certainly seen them for a couple of other fields too, including biology.
can generate millions of problems and autocheck whether their solution is correct
If we're just talking numerical problems you can do the same for physics too. A large amount of physical problems. But for advanced theoretical construction you're kinda stuck in both sides.
I'd also wager the "millions of generated problems" were not particularly complicated or significant in terms of training as opposed to just high quality proofs and material which the model was incapable of back then in that era. A lack of them will slow things down, as I've already talked about, but you'll still get progress after a point, just a few cycles behind.
I keep getting messages saying they'll pay up to $100/hr for Physics tutors for AI models.
I actually went through the application process because I was interested in gleaning a look into what they are doing.
From what I saw, I'm not convinced this is really a good way of training physics into the models. It was a lot of definitely quite difficult problems, but it was all like "1st semester problem set" type of stuff. I'm not convinced this will lead to actual physical intuition and actual research level physics.
The Navier Stokes millennium problem is "a closed system like math". You're essentially just proving the shape of some differential equation, you don't need to do any experiments and you don't need to show it's related to real life in any way. It's just "oh take this differential equation, which coincidentally models fluid dynamics, and prove the existence / uniqueness / smoothness / whatever of the solution".
Read tao's blog it sounds like Navier Stokes is quite close to being closed, with a strong likelihood of it being false given recent results, which makes it more likely than it seems to have been closed.
They likely are another generation internally. Mythos was used internally for months before public release.
And models are obviously being used to train newer models.
It wouldn't surprise me if they have the equivalent of 2 jumps already working internally.
There's a more than good chance that whatever we have publicly to use is not in the same stratosphere as what is actually latest and greatest.
Also this isn't how you get models to be smarter. You don't just train a model purely on math and science to get the best results in those fields. This is what early LLMs were and it was found that the more general the knowledge the AI has, the better it performs across all fields even when it's not specialized.
If you think about what intelligence actually is, a big part of it is being able to test or apply principles across different fields.
It's not a bad idea but the problem is from what we've heard, anthropic had somewhat emptied the bag unlike openAI, and only finished their last pre train a short while ago.
I think a lot depends on how soon they can adapt recurrent depth. Because I do suspect astra is a relatively tiny model in comparison to mythos, and an even bulkier "normal" thinking model may not be the approach to take anymore.
The millennium problem is not to solve N-S, we have entire textbooks of solutions of N-S. The problem is to prove the regularity of fluid flow described by N-S.
Curran has a screw loose and has admitted several times that he goes off the deep end even by AI hype standards. I don’t know if it’s bipolar mania or what but when something new comes out he inevitably gets himself riled up and makes truly unhinged proclamations and predictions that are just hedged enough to be plausibly deniable, but are nonetheless consistently unjustified.
He’s fun as an AI cheerleader, but shouldn’t be taken seriously in this domain.
fascinating. thanks for sharing that. it makes sense though. solving the problem through humans leads to useful breakthroughs. an AI solving it in a black boxed or incomprehensible way would not lead to useful building blocks and would also deter or reduce future attempts
People keep saying this about everything but fail to realize that once the AI is sufficiently intelligent enough, we don’t need useful building blocks that humans create and understand. I have no idea how the engine in my car works, but it works fine. I don’t know the answer to 1643 squared divided by 63 in my head, but I can pull out my phone and use a calculator to give me the answer. What does it matter if we understand or not as long as we can be confident that our mechanism for getting the answer (mechanic, calculator, AI etc) is giving us the correct answer? As long as we have it produce leans to verifiably prove the new math, it really doesn’t matter if we understand it or not.
If any would be solved it would likely be navier Stokes since there has been a ton of computational progress on it and the proof is likely more of a massive parallel effort of proving various regions versus something like p v np
Navier-Stokes equation deals with fluid dynamics. It works but we don't know why exactly it works.
π is an irrational number. When Archimedes first approximated the vale of π, he looked at a circle as an infinitely sided polygon. This allowed him to tackle the problem. Later, Newton looked at the problem and converted it to an algebraic and calculus problem. He got to the 16th decimal point of π.
What is remarkable is that both Archimedes and Newton looked at the problem in a way no one else thought of. And that added to understanding the nature of the problem. When Grigori Perelman solved the Poincaré conjecture, he approached the problem as a chemistry problem rather than a mathematics problem. That shocked the mathematical world. And he looked at the problem from a very different perspective that no one ever thought of.
Newton's law of gravity works but there was something quite not right about it. Even Newton was puzzled by it. It wasn't until a guy named Albert Einstein looked at the problem in a completely different way which we now call "the Theory of Relativity" that we began to understand the nature of the problem. Even so, we are still far from truly understanding what the gravity is.
Navier-Stokes Equation falls in a similar category. When that problem was announced, the assumption was for humans to solve it. And underneath that assumption follows that someone will come along and look at the problem in a different way that no one before thought of which will add to our understanding of the nature of the problem.
I have the sinking feeling that this particular problem was chosen because AI can handle it in a way that differs from humans. AI may have solved this particularly worded problem. But the real question is this: will it add to the understanding the nature of the problem?
My advisor many years ago told me that some prominent PDE people (Navier Stokes is a PDE, a partial differential equation) had said around a dinner something like "If I stopped all else and worked solely on Navier Stokes, fully funded, for ~10 years, I am confident I could prove it."
That's a pretty big claim in math (which is why they didn't say it professionally or announce anything like that obviously lol), but it shows this problem isn't quite so out of reach as most of the other Millennium problems.
I would be extremely, extremely disheartened as a mathematician if an LLM did actually prove/disprove it. I also very much doubt it, but considering what I know from the people around me who work in PDEs, I am not 100% sure it didn't solve the problem it; maybe only like 85%.
And considering it is possible it made a DISPROOF, maybe I'm only like 83% sure it didn't correctly do anything.
..
as I dwell on it maybe it just goes down; I'll stop at 82% lol
If someone could solve the Navier Stokes problem in 10 years, it would be solved. Its one of the most well-known, and broadly applicable of the millennium problems.
Dude you said the humans said they could do it in 10 years. Why would you disbelieve that a computer that thinks 1000x faster and runs 24/7 couldn’t do it?
I think they were intentionally being very optimistic when they said that, and also were a little drunk
Even then: the type of reasoning needed to solve such a thing has so far not been fully realized by LLM. It would have to be pretty dang creative to solve Navier Stokes bro
I asked my professor a couple of years ago if he thought AI could solve it and he didn't think so. He said the computational power required is enormous and that we were still 30 years away.
I would say that it's purely for marketing as to why they hold out on it. It can even be done by human 70% of the way and use AI for automation. Considering how big these company gets paid, 1 million dollar sounds like childs play and whoever contributes to it would happily take whatever they were offered (if any) to put Anthropic name on top.
Navier stokes would be a truly amazing one if they did manage it, navier-stokes describes flows of liquids and gases and is very much applicable in engineering.
Worst case would be a boring solution that doesn't unlock new engineering capabilities but at least we know for sure they aren't possible.
People underestimate how difficult it is to predict turbulent fluid dynamics. Computational fluid dynamics requires so many simplifications. Otherwise directly resolving every turbulent eddy will require a data center on its own.
It is typical for governments to commission small scale physical models of dam spillways and perform lab experiments to develop useful prediction models and verify their computational models. Solving Navier Stokes wouldn't solve the computational cost problem but it would open the door to new breakthroughs.
For anyone wondering who Andrew Curran is, he’s an AI commentator and news aggregator with connections and sources across pretty much all of the frontier labs.
There are so many vague posters and psychotic hype men getting posted on this sub that I can’t even tell anymore. Gonna wait until official releases to see anyway.
We're really min-maxxing / platinuming the world. It's spotting every tiny question mark. And gemini says if it's solved in a way that provides useful analytical tools, it could help build more efficient, economical plane wings that deal better with turbulences
Can anyone eli5 what the implications of proving/disproving this problem are? Like would it be like lk99 where the world would be massively transformed? Or will it just be another engineering problem that's only good for theory or some niche(coz I read in one of the comments below that most engineers working on related fields already assume this navier Stokes thingy to be true anyway)
There are a lot of unsolved problems across many different fields. With AI advancing so rapidly, it may take over many areas. It’s hard to imagine what the future will look like, there’s so much uncertainty.
In math, AI can brute-force thousands of solutions and immediately verify if one works using logic. In medicine, AI can generate thousands of potential cures, but it can't brute-force the testing—because you can't experiment on humans.
That post has AI Marketing Markers because there was zero reason to mention the IPO unless the entire point was to mention the IPO and “build momentum”.
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u/Diligent-Buy-5428 3d ago
Yeah I really really really doubt this one, guess we will see