As you may have heard, Open AI claimed to have solved the Navri-Stokes problem https://openai.com/index/navier-stokes-solution/

However, there are multiple discussions around that claim. For now, the full solution has not been presented so there are a few uncertainties regarding the validity of the claim:

Two weeks ago OpenAI claimed a solution to one of the biggest open problems in math—the Navier-Stokes problem—an achievement worth a $1-million prize from the Clay Mathematics Institute. The proof ignited a powder keg of concern over artificial intelligence companies’ race to disrupt the subject.

But with the dust still far from settled, a different controversy is emerging: Did OpenAI even solve the right Navier-Stokes problem?

Generated by an internal large language model (LLM), OpenAI’s proof relies on an approach that many experts find unnatural. It solves a variant of the problem that mathematicians say is disconnected from reality and thus less interesting. In a sense, the LLM found and exploited a loophole in the framing of the question.

“The most important problem is unsolved,” says Luis Silvestre, a mathematician at the University of Chicago. “The Clay problem is settled, but the main problem for the Navier-Stokes equations is not.”

There is also another controversy:

Over the course of August, Dr. Buckmaster and another mathematician had made significant progress on a longstanding puzzle involving the Navier-Stokes equations, which describe the flow of fluids like water and air.

But then OpenAI, the artificial intelligence behemoth, jumped into the fray.

Rumors that their archrival, Anthropic, had solved one or two of math’s “Millennium Prize problems” spurred OpenAI leaders to tackle the same list of problems, which offers $1 million for the first proof that passes a rigorous review.

The list includes the Navier-Stokes puzzle, and in less than a week, OpenAI beat Dr. Buckmaster and all of the world’s other human mathematicians, announcing on Tuesday that it had found the answer.

Over the past couple of days, a firestorm has ignited across the social media universe: perceived threats, dangled prize money and suggestions that OpenAI’s A.I. agents had somehow found and incorporated Dr. Buckmaster’s recent work into the company’s proof.

On Wednesday evening, in response to questions from The Times, OpenAI said in a statement that it was “categorically” impossible for its A.I. system to have been influenced by anything Dr. Buckmaster had done in the past two months.

https://www.nytimes.com/2026/09/10/science/tristan-buckmaster-openai-math-navier-stokes.html?unlocked_article_code=1.AFE.CA12.DVpcZwQn9Yl6&smid=url-share

That opened up a completely different question. Let's assume that a mathematical problem is solved without human input but at high cost (the estimated computation time was valued at around 15 mio) what is ultimately worth?

In truth, it was precisely the opposite: the clearest evidence yet that generative A.I., rather than aiding scientific progress, may be thwarting it.

For starters, two mathematicians who had been trying for months to reach their own solution to the famous problem say the new proof appears to have ripped off their unpublished work, which also used an OpenAI model. The company, after first equivocating, quickly changed its tune: It said the model it used for the solution had not looked at any of the prompts that the lead mathematician had entered into the OpenAI model within the last two months. Whatever the case, to get to the finish line in so little time, OpenAI had deployed an advanced model not available to the public, using computing power that would have cost an outsider an estimated $15 million. How could regular researchers compete with a secret tool that has the power to steal your work and mint its own money?

Flashy finishes like this don’t necessarily contribute to mathematical knowledge. Terence Tao, perhaps the most prominent mathematician of his generation, is generally positive about A.I. and math — so positive that he was featured in one of OpenAI’s ads. But contemplating the possibility of Navier-Stokes being solved by A.I., he wrote a long post explaining that “in most cases in pure mathematics, the problems are posed not because we desperately want the solution to these problems in and of themselves.” Instead, mathematicians want to see all the work that goes into achieving the solution — all the not-quite-right hypotheses that got adjusted this way or that, all the seemingly dead ends that “in fact end up being highly instructive in the nature of their failure.”

I also like the final point made in the last paragraph here:

Many scientific fields are already drowning in plausible-seeming but unverified papers generated with significant help from a large language model. What’s slop and what’s not? Is there an actual gem amid the gazillion new hypotheses? It’s becoming increasingly impossible to tell. Just last week, the editor in chief of Arxiv, a site where researchers can share papers that have not yet been peer reviewed, proposed implementing oral exams for authors who submit work, to screen out A.I.-generated junk. The scientific conversation cannot survive the deluge.

If OpenAI and Anthropic really want to help advance science, there is one obvious way they can do so: They can spend a tiny fraction of their vast fortunes to fund the kind of humble but essential research that’s increasingly starved for resources.

Often, AI compute time is seen as free, but it ultimately is expensive. And I wonder whether 15 mio in form of grants to further mathematical research group would have yielded more insights, for example. I also see parallels in the use of AI in biological research where (as also mentioned in the article) the actual work starts after hypothesis generation.

https://www.nytimes.com/2026/09/22/opinion/artificial-intelligence-ai-danger.html?unlocked_article_code=1.DVE.uGAZ._2SuQ4ojyQES&smid=url-share

1 hour ago, CharonY said:

“in most cases in pure mathematics, the problems are posed not because we desperately want the solution to these problems in and of themselves.” Instead, mathematicians want to see all the work that goes into achieving the solution — all the not-quite-right hypotheses that got adjusted this way or that, all the seemingly dead ends that “in fact end up being highly instructive in the nature of their failure.”

I've said it before, but according to a quote I've seen, a neural network has the second-best solution to any problem.

1 hour ago, CharonY said:

As you may have heard, Open AI claimed to have solved the Navri-Stokes problem https://openai.com/index/navier-stokes-solution/

Okay, I'm going to bite here. Somebody has offered a possible solution to (AN IDEALISED CASE OF ONE OF THE) Navier-Stokes (SMOOTHNESS) problems. Please forgive the capitalisations but without the extra words, some innocent fellow might be lead into believing that we were referring to a general solution to the Navier-Stokes equations.

In context, the smoothness problem is akin to the fractal coastline problem: prove that triangulation of the mean HWM length of the British coatline tends to infinity for a sufficiently short measuring stick. Solving this doesn't mean you've solved Earth Science in general.

Specifically, can the Navier–Stokes equations for a three-dimensional incompressible fluid with constant density develop a “singularity,” even when the motion starts smoothly?

In which universe are we supposed to find an incompressible fluid?

The maths might be interesting... useful even. But of any use to those for whom Navier-Stokes are a tool for use in the real world? Doubt it.

Edited September 23Sep 23 by sethoflagos

Seems to me, that if Dr. Buckmaster had published any of his work, any AI with internet access would be able to make use of it.
And I really don't think OpenAI included crediting Dr. Buckmaster's work in its 'footnotes'.

1 hour ago, sethoflagos said:

Specifically, can the Navier–Stokes equations for a three-dimensional incompressible fluid with constant density develop a “singularity,”

Sorry not you, but your link.
Anyway, whenever 'singularities' arise, we know that our model is no longer effective under those conditions.
Does OpenAI know this ???

1 hour ago, sethoflagos said:

Okay, I'm going to bite here. Somebody has offered a possible solution to (AN IDEALISED CASE OF ONE OF THE) Navier-Stokes (SMOOTHNESS) problems. Please forgive the capitalisations but without the extra words, some innocent fellow might be lead into believing that we were referring to a general solution to the Navier-Stokes equations.

Absolutely fair.

2 hours ago, sethoflagos said:

The maths might be interesting... useful even. But of any use to those for whom Navier-Stokes are a tool for use in the real world? Doubt it.

I think that goes towards the argument made in the article. From a mathematics rather then physics perspective, developing and approaching questions, even if they do not have any real-world counterpart can be relevant for the field itself, but only if the work is being demonstrated and can be read and understood by humans.

28 minutes ago, MigL said:

And I really don't think OpenAI included crediting Dr. Buckmaster's work in its 'footnotes'.

I think OpenAI offered to publish with Buckmaster as author. However, there was a second mathematician (Alpoege) who worked with Buckmaster which they didn't want to have on the paper. The reason was because Alpoege works for Anthropic. In addition to the maths and the issues of potential negative AI influence on science we also can add conflict of interest and the perhaps the negative impact of companies on scientific integrity....

1 hour ago, MigL said:

Anyway, whenever 'singularities' arise, we know that our model is no longer effective under those conditions.
Does OpenAI know this ???

Every 1st year Chem Eng student learns the known scope of applicability of the N-S equations: I'm sure it's written down formally somewhere (therefore accessible to LLMs), but from the dusty recesses of my gut recollectiom:

When system characteristic dimension approaches the order of magnitude of particle mean free path, the continuum approximation of N-S becomes invalid, and the system is better analysed via the domain of statistical mechanics (yuk).

Okay, found one version of it from MIT at The Limits of Navier-Stokes theory and kinetic extensions for describing small-scale gaseous hydrodynamics

We focus on the modeling challenges arising from the breakdown of the Navier-Stokes description, observed when characteristic lengthscales become of the order of, or smaller than, the molecular mean free path.

Near enough.

OpenAI apparently released some more stuff. What I wonder is how much of that effort hinged on human input and how much it cost. The Navier-Stokes work was, IIRC in the order of 15 million and I wonder if AI was strictly necessary or whether a group with other methods (including perhaps cheaper and lower level AI) could have advanced it similarly on that budget.

https://www.nytimes.com/2026/10/08/science/mathematicians-respond-openai-release.html?unlocked_article_code=1.HVE.VzAG.bgDC5hrDf8cb&smid=url-share

The NYT link is behind a paywall.

The 'unlocked' is an advertising fake, demanding my email.

Hmm it was supposed to be a gift article. Anyway, not a ton of details:

“If a human did this, it would be an instant Fields Medal, no questions asked.”

That’s how Alex Kontorovich, chair of the department of mathematics at Rutgers, responded to an A.I.-generated proof that OpenAI, the artificial intelligence giant, released on Tuesday.

And that single proof — worthy, in Dr. Kontorovich’s assessment on social media, of arguably the highest honor in the field of mathematics — was just one among more than 350 findings that the firm released that day, all at once.
Not all of them were quite so remarkable. But Martin Bridson, a mathematician at Oxford and the president of the Clay Mathematics Institute, which established the coveted Millennium Prize Problems, called OpenAI’s release “breathtaking.”

“Until very recently, it would have been impossible to imagine that the frontiers of mathematics could move so far in one day,” Dr. Bridson said.

If there had been any doubt that the leaps achieved by artificial intelligence this year would transform the discipline of mathematics, the tens of thousands of pages let loose on Tuesday have put that doubt to rest. The puzzles apparently conquered by OpenAI include crucial open questions in nearly every subfield of math, including algebra, number theory, theoretical computer science, mathematical logic and topology.
None of the proofs fully resolve any of the five remaining Millennium Prize Problems, which were designated at the turn of the century as a way of celebrating mathematics and its vast frontier of mysteries. But the trove of findings seemed to include results that make tangential or related progress on all of them. Perhaps the most significant result — the one Dr. Kontorovich was responding to — was a proof of the “quasi-Riemann hypothesis,” a conjecture related to the best known of the Millennium problems, the Riemann hypothesis, and a possible steppingstone of sorts to its resolution.

Ken Ono, a professor at the University of Virginia and the founding mathematician at Axiom Math, a math-focused artificial intelligence firm, wasn’t sure whether the quasi-Riemann solution would be a springboard for solving the original hypothesis. But he said that, as is expected with the original Riemann hypothesis, an encyclopedia of consequences would flow from this result, providing rich fodder for future breakthroughs.

“I imagine many mathematicians, like me, felt like they got punched in the stomach yesterday,” Dr. Ono said of Tuesday’s turmoil. “But that feeling is quickly giving way to excitement and determination. For those working to push mathematics forward, this is a challenge and an invitation to aim higher.”

(As of Thursday, OpenAI had made a number of updates to the public online repository; withdrawing three papers and making fixes to several others. A spokeswoman said: “Where errors are identified, we will work to correct them promptly and withdraw papers if no fixes can be found. As with other research manuscripts, this is an iterative process.”)

For those not familiar with Navier-Stokes, they are a set of equations describing fluid motion.
In the simplest terms, they are an application of Newton's second law of motion to a fluid, taking into consideration all internal and external forces acting on that fluid.
Further assumptions allow more simplifications, by considering only an incompressible fluid with constant density/viscosity, as this video derives

The next video will attempt to demonstrate various methods of solving Navier-Stokes equations.
I'll take understanding ( even if I can't solve ) over AI solutions any day ...

Here is an interview with the mathematician Terence Tao on this matter for a bit of perspective from that side. The link is a NYT gift link, but it is quite possible that they changed their system requiring an account: https://www.nytimes.com/2026/10/16/us/math-ai-terence-tao.html?unlocked_article_code=1.JFE.2wk5.5qGqYHZf1tLe&smid=url-share

OpenAI’s releases have now left him feeling used. “As recently as a year ago, I had a pretty good relationship with many employees at OpenAI,” he said in a recent phone conversation.

“A lot of people I interacted with shared a very common vision of how A.I. should be used — you know, to empower scientists.” But the rush to publish new findings, he said, shows their true purpose. They “used math for what they needed, which was to evaluate their models,” he said. “At this point, I don’t see a way to constructively engage.”

His anger was clear in a post on the widely read math blog Proofs and Prompts after OpenAI’s Oct. 6 release. “I am horrified by the many person-years” of research, particularly by young mathematicians, “being casually disrupted or destroyed,” he wrote.

“Much as one cannot unhear a movie spoiler or a crossword clue, one cannot explore a problem as profitably and richly once one is aware of an existing solution.” He mourned, he added, “the path not taken, and the opportunities lost in the frantic race to develop this technology.”

[...]

Theoretical, or “pure” math — the math of Dr. Tao — is loved as much for its art as its science. Its primary goal is to add to the understanding of the vastness of math itself. Unlike applied mathematicians, who have generally embraced the power of A.I. to help them solve practical problems in engineering, finance, weather forecasting and other sciences, pure mathematicians study shapes, structures, numbers and abstract concepts.

Still, pure math ultimately has led to, among many other advances, the creation of the internet, microchips, GPS, cybersecurity, encryption, CT scans and other medical procedures. In 2006 Dr. Tao and three partners created the mathematical theory of compressed sensing, which sped up the scan times of MRIs. Today MRIs can produce real-time images of the beating heart, and many children can be scanned without sedation.

[...]

Dr. Tao was not impressed with the writing in OpenAI’s proof of the Navier-Stokes problem. “It is technically readable,” he said. “But no expert I’ve talked to has actually understood the whole argument.” He said the writing is better in the latest release, though he called it a “mixed bag.” Some papers appropriately cited his work, he said, and seemed to have advanced it in a way he thought was plausible. Still, “there is no author that I can have a real conversation with about the results.”

Dr. Tao was aware last month that OpenAI was working on the Oct. 6 release and at the time called the coming results “toxic waste.” Now he calls them “maybe treated waste.”

[...]

To Dr. Tao, the next few years will require the reinvention of math, or at least of math education. The Harvard report recommended, essentially, that math become more human-centric: Doctorate degrees should not be rewarded primarily on the text of a dissertation, as they are now, but more on a rigorous in-person defense. Students should regularly show professors they can speak fluently about their work.

Dr. Tao went further last week when he presented what he sees as the future of math in a lecture at the California Institute of Technology. Despite OpenAI’s releases, he said, most problems at the frontiers of math are unsolved, math remains infinite and the difficulty of problems is therefore boundless. Since A.I. has given mathematicians the ability to collect data on such a vast scale, they will increasingly work in large groups collaborating on once-unimagined projects.

It is interesting that one of the impacts of AI in mathematics that it might become more similar to other, more technical fields, which rely more on collaborations already.

On 9/23/2026 at 9:25 PM, sethoflagos said:

In which universe are we supposed to find an incompressible fluid?

The maths might be interesting... useful even. But of any use to those for whom Navier-Stokes are a tool for use in the real world? Doubt it.

I quite agree with what you said here. There's also the matter of whether the external force can be seen as realistic at all for the case giving rise to singularities (something I have not considered in any detail). Furthermore, as it is an equation for a velocity field and we know celerities greater than the speed of light are forbidden, Navier-Stokes cannot be a fundamental equation of physics the way other microscopic equations are suspected to be.

I think it's more mathematicians, or mathematical physicists, that consider the NS problem a holy grail of science.

11 hours ago, joigus said:

There's also the matter of whether the external force can be seen as realistic at all for the case giving rise to singularities

Reality is always real.
Our modelling sometimes, does not fit that reality.

11 hours ago, joigus said:

it is an equation for a velocity field and we know velocities greater than the speed of light are forbidden, Navier-Stokes cannot be a fundamental equation of physics

Oh, good Lord !
Relativistic Navier-Stokes Equations ?!?!

4 hours ago, MigL said:

Reality is always real.
Our modelling sometimes, does not fit that reality.

Oh, good Lord !
Relativistic Navier-Stokes Equations ?!?!

I'm not asking for that. But don't forget the Navier-Stokes equations are diff. eqs. for a velocity field. And the conundrum was "are there singular solutions?" Meaning: "do they blow up in finite time?" So how could the velocity components becoming infinite have any fundamental significance? I say this knowing full well that some people around here, good people, do not like the word "fundamental".

I'm glad we're back.

No kidding, there are theoretical frameworks for highly relativistic fluids in astrophysics. When the model doesn't work anymore, one is forced to rethink the model. It's a completely different outlook to that of mathematicians. For them the model is the thing.

20 hours ago, joigus said:

I think it's more mathematicians, or mathematical physicists, that consider the NS problem a holy grail of science.

For me, it's the sheer chutzpah of calling it THE Navier-Stokes problem, as if it were even in the same ballpark of difficulty (read impossibility) of a true general solution.

20 hours ago, joigus said:

Furthermore, as it is an equation for a velocity field and we know celerities greater than the speed of light are forbidden, Navier-Stokes cannot be a fundamental equation of physics

They are nevertheless very useful in a number of mundane applications.

But since you mention celerity, let me clarify my earlier comment about the (supernatural) 'incompressible fluid' assumption.

In general application, the partial derivative of pressure with density is often significant and has to be dealt with. For an ideal gas under isothermal conditions it is trivially equal to RT, an isentropic process, kRT, which is also the square of the speed of sound. The incompressibility assumption sets this value to infinity, resulting in propagation of forces not just superluminarly but instantaneously.

What, pray, is the value of demonstrating a singularity in a solution when one has artificially introduced one as a core simplifying assumption in the problem set up? Is this not a prima facie case of begging the question (in the traditional sense of circular reasoning)?

5 hours ago, joigus said:

No kidding, there are theoretical frameworks for highly relativistic fluids in astrophysics. When the model doesn't work anymore, one is forced to rethink the model. It's a completely different outlook to that of mathematicians. For them the model is the thing.

Absolutely! NS is just a starting point. In practical usage: simultaneous solution of a continuity equation is invariably involved; the bulk modulus related terms discussed above arise through importing an EoS; non-newtonian viscous behaviour requires a rehashing of the diffusion terms. Adding Lorentz invariance would just be another in a long line of add-on modifications to suit the particular application.

Edited October 18Oct 18 by sethoflagos
typo

17 hours ago, sethoflagos said:

For me, it's the sheer chutzpah of calling it THE Navier-Stokes problem, as if it were even in the same ballpark of difficulty (read impossibility) of a true general solution.

Agreed. Although there isn't such a thing as a general solution. In the mathematical theory of ODEs, eg, not even linear equations have known solutions for any inhomogeneous term (analogous to applied external force). They can be systematically found for a catalogue of simple known functions. Let alone non-linear systems of eqs. with arbitrary applied force. I've spent years wondering myself 'why is this problem so crucial?'

I have learned something from this thread ...

@joigus used the word 'celerity' in his post, and when I quoted him, I assumed the auto-correct on his phone had over-reacted, and 'corrected' celerity to velocity.
( I thought celerity might have something to do with being celery )
Turns out celerity is a term meaning rapidity or swiftness of motion/action.

My apologies for assuming error of others, instead of my ignorance.
( learning is a humbling, but joyful, experience )

36 minutes ago, MigL said:

I have learned something from this thread ...

@joigus used the word 'celerity' in his post, and when I quoted him, I assumed the auto-correct on his phone had over-reacted, and 'corrected' celerity to velocity.
( I thought celerity might have something to do with being celery )
Turns out celerity is a term meaning rapidity or swiftness of motion/action.

My apologies for assuming error of others, instead of my ignorance.
( learning is a humbling, but joyful, experience )

It's essentially the modulus of velocity. I'm a stickler for precise language, sometimes to unreasonable lengths. Because kinetic energy depends on (modulus of velocity)², I figured that was the proper term relevant there. I imagine v² having an enormous value would have presumably many physically relevant consequences (dissipation, temperature, and what not) that could make the standard NS mathematical model really not that terribly important in terms of the physics.

I'm not anywhere close to being an expert on all this though.

3 hours ago, joigus said:

Agreed. Although there isn't such a thing as a general solution. In the mathematical theory of ODEs, eg, not even linear equations have known solutions for any inhomogeneous term (analogous to applied external force). They can be systematically found for a catalogue of simple known functions. Let alone non-linear systems of eqs. with arbitrary applied force. I've spent years wondering myself 'why is this problem so crucial?'

A differential equation is considered solved if it is reduced to a problem of integration. That's no consolation if the difficulty is with the integration. For example:

[math]\displaystyle \dfrac{d^2y}{dx^2} = ky^2[/math]

[math]\displaystyle \dfrac{dy}{dx} \dfrac{d^2y}{dx^2} = ky^2 \dfrac{dy}{dx}[/math]

[math]\displaystyle \int \dfrac{dy}{dx} \dfrac{d}{dx}\left(\dfrac{dy}{dx}\right)\ dx = \int ky^2 \dfrac{dy}{dx}\ dx[/math]

[math]\displaystyle \int \dfrac{dy}{dx}\ d\left(\dfrac{dy}{dx}\right) = \int ky^2\ dy[/math]

[math]\displaystyle \dfrac{1}{2} \left(\dfrac{dy}{dx}\right)^2 = \dfrac{k}{3} y^3 + C[/math]

[math]\displaystyle \dfrac{dy}{dx} = \sqrt{\dfrac{2k}{3} y^3 + 2C} = \sqrt{\dfrac{2k}{3}} \sqrt{y^3 + A}[/math]

[math]\displaystyle \dfrac{1}{\sqrt{y^3 + A}} \dfrac{dy}{dx} = \sqrt{\dfrac{2k}{3}}[/math]

[math]\displaystyle \int \dfrac{dy}{\sqrt{y^3 + A}} = \sqrt{\dfrac{2k}{3}}\ x + B[/math]

In this case, the original differential equation has been reduced to a problem of integration. However, the integration appears to be some form of elliptic integral. Actually, one could argue that a second-order differential equation be considered solved if it is reduced to a first-order differential equation. For the above equation, that is indeed straightforward. Interestingly, it is the arbitrary constant from the first integration that creates the difficulty for the second integration.

Edited October 18Oct 18 by KJW

4 hours ago, joigus said:

Agreed. Although there isn't such a thing as a general solution. In the mathematical theory of ODEs, eg, not even linear equations have known solutions for any inhomogeneous term (analogous to applied external force).

I'd hoped to convey this sort of idea via use of:

21 hours ago, sethoflagos said:

(read impossibility)

53 minutes ago, KJW said:

A differential equation is considered solved if it is reduced to a problem of integration. That's no consolation if the difficulty is with the integration.

Computer science people would probably consider a differential equation solved if existence and uniqueness is satisfied, as they can always solve it by feeding it to the software. There's always the question of when experts consider a problem to be of sufficient entity.

1 hour ago, sethoflagos said:

I'd hoped to convey this sort of idea via use of:

Yes, sorry. You did.

2 hours ago, KJW said:

one could argue that a second-order differential equation be considered solved if it is reduced to a first-order differential equation

That's usually done by techniques like separation of variables, which usually involves making some 'simplifying' assumptions about the system in question.
The solution to the partial differential Hydrogen atom comes to mind, as two of the variables are easily separated due to spherical symmetry considerations.

This is essentially what is being discussed; exact solutions to N-S may only be available if simplifying assumptions can be made.
Otherwise, it's numerical methods by computer.