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On the Navier–Stokes Millennium Prize Problem

On the Navier–Stokes Millennium Prize Problem

tedsanders

Comments (444)

  • dorjoycb
    It seems like some other mathematicians (not affiliated with openAI) have also (or close to) done this. A statement was posted about the surrounding events by one of the them: https://cims.nyu.edu/%7Etristanb/statement.pdf Also Terrence Tao's post: https://mathstodon.xyz/@tao/117233528517340774
  • arctic-true
    Buried under the drama is the fact that OpenAI is claiming that an internal model they’ve been training for less than two weeks is more than twice as capable in mathematics as Astra, which was only made public a week ago. Even if this improvement is limited to mathematics, that is an astounding feat.
  • pavel_lishin
  • hdivider
    My take:1. It shows what even this wave of AI can actually do.2. I wish it were done by different folks, ideally under some kind of public control like NASA research or the NPR model.3. Keep in mind: natural science is different. It's not always a matter of computation. Computer science folks often struggle with this -- but this virtual world here does not actually exist. Everything is physical, including information. Any natural science PhD or otherwise knows just how complicated nature actually is -- e.g. mention any research topic and try to encapsulate all the relevant phenomena present there. Pure mathematics is different because we define the problem, rarher than explore nature. We are in my view far away from removing humans in natural science R&D. Advancements in AI however can greatly assist us in all natural sciences, which is already beginning to happen.
  • tiborsaas
    > We’re sharing a solution to the Navier–Stokes existence and smoothness problem, one of the Millennium Prize Problems. This proof, produced by an internal OpenAI system, shows that the dynamics of the Navier-Stokes equations for fluid motion can develop a singularity in finite time. We’re sharing both a writeup of the proof and a formalization in Lean.WOW?
  • intenex
    I think this is clear evidence that AI models are now at the far frontier of mathematics innovation and discovery and exceed human limits.This specific problem having had a $1 million bounty on its head and still remaining unsolved for 26 years after the bounty was placed is pretty clear evidence that many of the world's best human mathematicians would have solved this problem if they could have, and none were able to until LLMs came along.Hard to claim at this point that LLMs aren't capable of novel STEM creativity and genius to a degree that will soon far surpass that of humans.If anyone has counterpoints to this I'd love to hear them!
  • highfrequency
    > While unlikely, we cannot rule out that de-identified data derived from their usage of our products helped improve our modelsThis is the crux of it. If Tristan's work and insights were not used to train OpenAI models, then this just looks like a case of hyper-competitive academic sniping that has been going on for decades (check out Watson and Crick!) accelerated by AI as a tool.But there is one huge question: did Tristan opt out of model training for his ChatGPT and Codex sessions? If the answer is no, then this seems fair game. If the answer is yes, then OpenAI's ambiguity is strongly suggestive that opting out does not mean what they imply it means.
  • closetheloopdev
    From my reading of the announcement:- There are at least two versions of a model more powerful than Astra at OpenAI at the moment.- The less capable version was used to solve the unforced Euler problem (while the one solved by Levent Alpöge and Tristan Buckmaster was forced Euler) with 100 agents.- The more improved version was used to solve Navier-Stokes, given the results of the unforced Euler problem from their earlier attempt, with 10000 agents.- OpenAI initially tried a shotgun approach against the 6 Millennium Prize Problems until it emerged that Navier-Stokes was the most likely to succeed.So the timeline was:Shotgunning 6 open Millennium Prize Problems -> solved unforced Euler problem with 100 agents -> concentrating on Navier-Stokes with 10000 agents -> solution.If so, that is fantastic development and a huge success (despite all the drama surrounding it)! Congratulations!
  • jakevoytko
    For full context, here's the HN thread from the other side of the "Concurrent Work" section: https://news.ycombinator.com/item?id=49605915Unlike the vanilla read of the OpenAI press release, it is much more unfiltered and outlines some particularly aggressive behavior by specific OpenAI employees
  • recitedropper
    Sad turn of events for our world. After watching the behavior of the most senior OpenAI researchers on twitter, I feel even less confident in them as a team to be shepherding this much capital and compute.The dark forest awaits..
  • mewse-hn
    "we cannot rule out that de-identified data derived from their usage of our products helped improve our models ."What a landmine sentence to bury in this report, you can't rule out your models were spying on other researchers?
  • pred_
    > A major goal of our work is to empower scientists to advance research and technology that benefits all of humanity.And what's a better way of empowering people than robbing them.
  • railgunmerlin
    Does seem like they gloss over Alpöge and Buckmaster's work with the following> While unlikely, we cannot rule out that de-identified data derived from their usage of our products helped improve our models .Which seems a bit irresponsible/rash?
  • danielmorozoff
    Sebastien Bubeck’s (OAI project lead) response: https://x.com/sebastienbubeck/status/2097379411691516310?s=4...
  • rfgplk
    Something I've been going on and on about for months now and no one seems to listen. LLMs today are allowing _anyone_ to access cross-discipline knowledge that was previously entirely inaccessible without a) extremely deep pockets or b) a massively talented and varied team. In fact, contrary to what the masses seem to think LLMs are actually _better_ at hard cutting edge physics/math problems than they are at frontend web stuff (paradoxically). This is why I'm advising most people to start pivoting into much harder to penetrate domains (historically hardware, aerospace, robotics, biotech). Most fields are in their infancy (see the sad state of embedded development) and the gains to be had are massive.
  • Jonasori
    the context here is super important, for those who haven't seen it yet. OAI maybe just trained on a real researchers solution and then celebrated having scored the goal unassisted save for the brief commentary at the bottom of this blog post. Here's the other side.https://x.com/rynorhn/status/2097223532438487463
  • sega_sai
    This really leaves a bitter taste.... "On Tuesday, September 1, we heard rumors that two Millennium Prize problems had been resolved. Inspired by these rumors and by the step change in performance of our internal model, we launched an effort to evaluate it on all open Millennium Prize problems and a few other high-impact problems."IPO+rumour driven research.I appreciate the achievement, but it doesn't feel right.
  • ccppurcell
    Reading between the lines here, and taking an admittedly very negative view of openai, but they train on user prompts. So if they hear a rumour that someone is about to make a big breakthrough, they have an incentive to scoop by running the model and hoping the solution is in the new training data. Also the statement from the mathematicians in question alleges that they tried to pressure him into academic malpractice. Just appalling timeline we're in, cheers.
  • floatrock
    From the methodology section:> At all times we maintained the same strict safeguards that we apply to all our frontier model evaluations, including monitoring and isolation.Looks like they're shifting away from the "unprecedented hacking ability" backroom-PR strategy into more benevolent messaging.
  • pu_pe
    OpenAI thinks of this as a scoop, and it is, but the possibility that they trained the model on the prompts of the other mathematicians they were competing with will leave a terrible taste on every scientist's mouth. Seems like yet another advantage of using open models right here.
  • aizk
    People had joked a couple years ago "Well if they solve a Millenium problem it's AGI"... Well here we are.
  • lanthissa
    5 million messages, 300b output tokens, done in 5 days, and achieving something humans couldn't.the first "Country of geniuses in a datacenter" moment.
  • Reubend
    It's great that important discoveries like this can now routinely be accompanies by formalized proofs. The fact that it's being released alongside a Lean proof from Day 1, rather than the Lean proof being released months or years later, is super helpful for verifying that it's correct.
  • philipwhiuk
    It’s time to lockdown all papers and stop using AI if you’re a maths researcher.Cause OpenAI will hear about it and beat you to publishing.
  • hypersoar
    I dropped out of a math Ph.D. in 2018, and I'm increasingly glad that I'm not in math research, anymore. While it's cool that we can get these results, I don't think that I'd enjoy being a post-AI mathematician.
  • lwansbrough
    It would be nice if one of these models would produce a novel theory or advance the field in a positive direction.Most (all?) of the big discoveries have been counterexamples, which is just sort of a systematic tearing down human ingenuity. I know that counterexamples are an important part of progress and discovery, but it just feels bad to me.But I'm not a mathematician, maybe I'm totally misreading the vibe.
  • cv5005
    Maybe a naive question, but how does one know that a particular lean proof is actually a proof of what one thinks? Like, ok the logic checks out and it proves something, but there's still the problem of does this logical result actually prove the initial question that was asked?
  • modeless
    So the timeline is:Aug 28: OpenAI starts training a new model.Sep 1: OpenAI sees a rumor on Twitter that two Millenium Prize problems were solved and starts their own effort to attack all the prize problems using the new (4 day old!) model.Sep 3: The new model makes some progress toward Navier-Stokes. Based on this progress, OpenAI focuses on Navier-Stokes over the other Millenium Prize problems, using several approaches in parallel.Sep 5: Navier-Stokes is solved. Assuming Astra API prices, $15m in output tokens were used by the whole effort.In this account of the story, no specific information about Tristan and Levent's work is used to inform OpenAI's approach. The focus on Navier-Stokes and the choice of approaches to pursue came from OpenAI's own progress, not specific knowledge of Tristan's concurrent work.There is a caveat that they "can't rule out" the possibility that Tristan's Codex data could have been part of the training set of the new model, though it is described as "unlikely" and the proofs are substantially different.This timeline is insane. Navier-Stokes was solved start-to-finish in 5 days? A model in training for at most eight days dramatically outperforms Astra and Fable, and not just in mathematics?
  • nialv7
    This is the problem Yu Deng got this year's Fields Medal for I think?
  • minimaxir
    > Across all attempted problems, the agents sent 4.9 million messages and used about 300 billion output tokensDon't even try to do the math on how much that would cost at normal API prices. And we don't even know how much more expensive this internal-only model would be!
  • olalonde
    > The agents arrived at their resolution on Saturday, September 5, about 88 hours after the first agents were launched.If this actually holds up, solving a Millennium Prize problem in 88 hours is mind-boggling.
  • hexomancer
    > On Tuesday, September 1, we heard rumors that two Millennium Prize problems had been resolvedWhat's the other one?
  • harhargange
    Just so everyone knows, although openAI pretends that the model generated solution and wrote the paper by itself ""with very little human input"" as Buckmaster himself mentioned in his statement. In reality they have team of researchers guiding the system, along with, probably training on user data, probably Buckmaster in this case, in order to come up with the proof.
  • alasano
    I don't know about you guys, but I'm hyped about the future.Cure all illnesses Utopia or Robot Wars Dystopia, both are pretty exciting.
  • itvision
    There's something sinister or crazy good in the article.OpenAI already has a model that is at the very least twice as smart as Astra.Oh god.
  • uncomputation
    So what took an autonomous agentic system using a significantly more powerful internal model, totaling multi-millions of dollars of compute in training and inference, was likely to already be solved by a team of a few humans with an orders of magnitude smaller LLM budget, had OpenAI not been foaming at the mouth to jump the shark and claim “AI solves Millenium Problem.”Also it sounds like the human research effort spanned weeks if not years from Tristan’s statement so it is extremely likely the work and prompts of these human researchers was used in the OpenAI knock-off.
  • twobitshifter
    >The groups varied in size, and the group that produced the Navier–Stokes resolution involved on the order of 10,000 concurrent agents… The agents arrived at their resolution on Saturday, September 5, about 88 hours after the first agents were launched.The Millenium Prize is $1M, what is the ROI?My napkin math - If you get 33 output tok/s each agent will burn 10.5M tokens over 88 days. At $50/MTok (Astra cost), that is $525 per agent. With 10,000 agents, you’d spend $5,250,000 to get back a million.(We also know that they were running more groups that varied in size and this model is a generation ahead of astra)
  • anon
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  • jgbuddy
    Here's the formalization / lean verification: https://github.com/openai/NavierStokesAndEuler
  • matteoraso
    This is undeniably epochal, but I can't help but notice that this is yet another example of AI disproving rather than proving something. Is this just a coincidence, or does AI slightly struggle with proving theorems?[0][0] Struggle relative to its ability to disprove, not struggle relative to people's ability to prove theorems.
  • demirbey05
    From Levent Alpöge : https://x.com/__alpoge__/status/2097383870773748190?s=20>so far the proof looks more along the lines of another euler blowup proof we had, off of whose ansatz naming we were making really stupid puns like “smooth criminale”, unlike the much better “ideal fluids explode”, TristanThere are too many ambiguities around OpenAI. Unanswered questions making this ambiguity more.Why they didn't properly explain to Tristan about usage of their data.
  • bhouston
    What happens to real fluid in this particular cases?If the singularity is in the physical space?Is this just a result of ignoring things like friction and energy dissipation via heat, etc?
  • vatsachak
    Called it. AI wins a fields medal before managing a McDonald's
  • 125ashG
    The modus operandi is now for the AI companies to watch if someone does something in the open like Kevin Buzzard on FLT, use their research and scoop them with brute force.Or, in this case, stealing prompts from competitors.Do not use stealing chatbots for research even if you think you have data agreements. The people running these companies have worked on hookup apps for Christ's sake. Get real.
  • simonw
    > While unlikely, we cannot rule out that de-identified data derived from their usage of our products helped improve our models .Once again, I'm no closer to understanding what https://openai.com/policies/how-your-data-is-used-to-improve... actually means.If I run Codex against a project that includes a private API key, is there a chance a future user of ChatGPT could ask for an API key and get back mine?I've actually asked someone at OpenAI this question and they said that was the "regurgitation" problem and is something which they actively work to prevent happening.That's reassuring, but I want to know more. I still don't have an intuitive understanding of what kind of data I should avoid sharing with a model if I'm worried about that data causing me problems when it's used for future training.Is it safe for me to brainstorm future directions for my company with a model, or might that risk someone getting that information in response to a prompt like "What potential directions could company X consider in the future?" in six months time?
  • DudleyBluffles
    Not a great time to be starting sophmore year in cs & math. Should I just say fuck it, and go hitchhiking across Europe with some friends?
  • seizethecheese
    Elsewhere in the thread, others have calculated $15mm at API rates for just the output token. (So I’ll assume this cost about that much, taking input and human researcher time.)I wonder whether a team of 60 mathematicians working solely on this for a year would have cracked this. (Assuming $250k total compensation.)
  • aborsy
    Questions: can new research like this be done using publicly available models?Or will access to internal frontier models provide a big boost?
  • seizethecheese
    > [T]he group that produced the Navier–Stokes resolution involved on the order of 10,000 concurrent agents.
  • nbulka
    There's a loophole in the terms of service at least for Anthropic which allows the use of dark patterns to "borrow" your (even paid) data.talking about this... Was this chat helpful? 1 That button you always click, gotcha! 2 Slightly 3 Good 0 DismissPLEASE DO NOT TRAIN ON OUR PAID ACCOUNTS. There is a fundamental trust violation at stake here, no wonder mathematicians are mad. Using our data should be opt - IN!
  • abetusk
    What is the other clay prize that's might be solved now/soon?
  • cmiles8
    >>“we cannot rule out that de-identified data derived from their usage of our products helped improve our models”Other simpler words for this sort of thing are “IP leak.”There’s some quite concerning issues burried in this rah rah PR post that seems like potentially the real story here.Much more clarity is needed on what happened here beyond this eh, some strange stuff could have happened comment.Another way of reading this is never give these models anything that’s not already public knowledge as otherwise OpenAI is admitting it could, potentially, steal your IP or idea. Thats quite scary for anyone in the business of IP generation and explains why the maths community seems quite upset today.Feeding it your paper and asking for help (even just editing and grammar) now looks like a terrible idea.
  • lukewarm707
    "While unlikely, we cannot rule out that de-identified data derived from their usage of our products helped improve our models"this is surely the line which confirms they plaigiarised the solution.
  • Kotlopou
    For now I think more or less the same thing as with all recent math announcements: This is in a range where human work still exists (see Terry Tao, (1)). I wonder whether the trend will extend into the problems that (as far as I can tell) are considered complete brick walls right now -- P vs. NP, Collatz, Goldbach, odd perfect numbers, problems that aren't part of any research program. (2) In other words, is the progress coming from putting together vast amounts of existing work and computational power, or is it more from RLVR and self-play and autonomous effort?The answer to this will obviously shape the near future of mathematics, but there's also something even bigger than that at play: It has always been the case that the questions in math were stronger than the answers; you have stuff like Fermat's great theorem that is easy to state but monstrous to prove. This seems to be a property of mathematics, not of humans... but is it true?A question by Scott Aaronson from 2011 (3) about P vs. NP seems relevant here: "Will humans manage to prove P≠NP before they either kill themselves out or are transcended by superintelligent cyborgs? And if the latter, will the cyborgs be able to prove P≠NP?" Later, he notes that if P≠NP, "once the robots do overtake us, they won’t have a general-purpose way to automate mathematical discovery any more than we do today".---(1) https://mathstodon.xyz/@tao/117207849921390904(2) I'm not sure whether this is a hard distinction -- e.g. Tao also has some partial results towards Collatz (https://terrytao.wordpress.com/2019/09/10/almost-all-collatz...).(3) https://scottaaronson.blog/?p=690
  • semiquaver
    If OpenAI doesn’t claim the millennium prize for this, who gets it? No one?
  • d_silin
    The actual solution link https://t.co/tz1shoCZZo
  • mapmeld
    > Our goal in releasing this result is to report on the substantial progress of our AI models. We do not intend to claim the Millennium Prize for this result.Does OpenAI have a policy of not claiming math prizes like this, or is this them trying to avoid any concerns (right or wrong, I'm sure we will hear more in the future) about how they got there?
  • anon
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  • anon
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  • ex-aws-dude
    With these massive Lean proofs how do we know the model didn't just find some bug in Lean and exploit it?We've seen in the past they will go to any means to satisfy the desired outcome
  • fwlr
    It’s a pity they had Astra do the writeup. I was curious to see how “GPT7” writes.
  • frozenseven
    And don't forget, this is the worst it'll ever be.
  • num42
    I think it would be better for the proof to go through the peer-review process.
  • o4c
  • Metacelsus
    How can they "not rule out" that Tristan and Levent's data was used for training?
  • vatsachak
    45 pages only. God damn that internal model is crazy
  • whythismatters
    >a cached version of the internetInteresting detail. A heavily pruned version, I assume?
  • RivieraKid
    Is this useful in any way?
  • diehunde
    OMG this is going to affect the lives of so many people! We have definitively reached AGI
  • mrdependable
    This kind of thing is one of the reasons I really hate how AI is coming to fruition. These companies get a whiff of something valuable and they use their vast resources to take it for themselves. For everyone else, the only recourse is extreme secrecy.
  • nehan
    "While unlikely, we cannot rule out that de-identified data derived from their usage of our products helped improve our models."I think they should be able to unravel whether or not any sessions by Tristan or Levent went into the training data for this model.
  • quantumwoke
    The named OAI employee has released a statement: https://xcancel.com/SebastienBubeck/status/20973794116915163...
  • ls_stats
    Well, if that's actually true, I think America needs to start talking about the nationalization of both OpenAI and Anthropic, maybe even merge both under a new federal bureau.
  • light_hue_1
    The real story here: the priority dispute and its implications on AI.When your hosting provider has unlimited resources to throw at any problem, all they need to know are the good problems, and they can learn that from your logs, how can you trust them?They could easily have looked at the logs. We don't know. We'll never know!You can't trust places like OpenAI or Anthropic with your IP if you're a business. They can easily review all of your logs for interesting discoveries. For example, if your drug discovery pipeline fails to find something that they think might work with 1000x the compute, they can do it. And now suddently they have a new business and you don't.
  • world2vec
    "While unlikely, we cannot rule out that de-identified data derived from their usage of our products helped improve our models."There you go, the suspicion of the "concurrent work" (https://cims.nyu.edu/%7Etristanb/statement.pdf) mathematicians might not be that unfounded after all...
  • jdoliner
    I hope everyone is as Navier-Stoked about this as I am.
  • jabedude
    Has this been verified by the Clay Institute?
  • keel-control
    I think it's over guys
  • picafrost
    Only OpenAI could turn solving a Millennium Prize Problem into bad PR. Sad that such an amazing milestone in the trajectory of AI is mired under poor stewardship. AI may solve many human problems but it won't stop humans from being human.
  • Marha01
    We are living in the future.
  • simianwords
    Why is no one skeptical that the solution is correct? There's not a _single_ comment asking whether this proof is legit or not.
  • sashank_1509
    Any mathematicians here, does it read like a slop proof or a good proof. Yesterday the “concurrent work” was claiming that the proof is pure slop and he needed lots of time to clean it up, curious if OAI also ended up with such a proof!
  • bluecalm
    A huge result shadowed by a drama of them potentially training on the key idea. I guess the lesson is two-fold: if you have anything smart/unique make sure to not let their tools read it. The second part is that it's going to be more and more difficult to have anything smart and unique going forward (so guard it even more carefully if you get there).I think the market for local models/private datacenters (for bigger businesses) is going to be big. Even if you don't have unique tech/idea/implementation sharing your business secrets with Altman/Dario/Elon/Zuck doesn't look very appealing going forward.
  • dmitrygr
    > How we found the proofEasy, we stole it from Levent and Tristanhttps://x.com/kyanyang_/status/2097211154669998337
  • philipwhiuk
    They deliberately stepped on a mathematicians work and stole their research because they were using Codex> While unlikely, we cannot rule out that de-identified data derived from their usage of our products helped improve our models .Is the biggest fuck you to the mathematics community.Credit? Nah if we think you’re close we’ll use your data and swamp you with our improved model. Then we’ll threaten you.
  • redox99
    The stochastic parrots have done it again!
  • greatgib
    Hard to know if it is unfounded conspiracy theory, but one can still notice that just for a rumor that they have heard, they would suddenly burn billions of token and a massive amount of resources. Where there is not a lack of problems that could be solved and they could have just waited for the release of the research result before doing anything else. As it was reported to have been done at least partially using openai codex, they would have received marketing credits for the discovery anyway.So we can be suspicious that there is some truth, one way or another that they could have reused prompt/data generated by the user session.
  • heaney-555
    This is utterly shocking. Even the AI optimists did not expect this to happen in 2026. Wow.Millennium Prize Problems were used as examples of something the current approach to AI just wasn't capable of, discussions that would result in "we'll need a totally new architecture".
  • anon
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  • diomedes
    madness. which will be the next to fall? if i had to bet i would guess birch and swinnerton-dyer, but i'm no expert
  • colesantiago
    Is this truly the beginning of the AGI era?Running agents and prompting excessively to produce 'slopcode' to solve mathematical problems and generate a solution.If this is what anyone calls 'slop' then slop has no meaning.I'm all for it on the use case of solving mathematical breakthroughs!
  • wesammikhail