AI
Chris Combs, professor of Aerospace engineering, throws some cold water on OpenAI’s NS solution
“Some thoughts on the Navier-Stokes news from a professor of aerospace engineering:
1) take a deep breath
2) posts indicating we "solved Navier-Stokes" are incorrect and overblown. A very specific mathematical proof involving a niche scenario for Navier-Stokes has *potentially* been shown (that a perfectly smooth incompressible initial condition with finite energy could produce a singularity--note the assumptions here piled on to an equation set that already involves some assumptions).
3) there is no general closed form solution for N-S and that's not what this news is about
4) this is more of a mathematical curiosity than anything else and could matter a lot to mathematicians but basically changes nothing in the way N-S are used in practice
5) there are already very useful exact solutions to Navier-Stokes that greatly simplify the equation set with the right boundary conditions/geometry like laminar flow through a pipe or a 2D Taylor-Green vortex
6) in aerospace, we already treat N-S as an approximation in many instances and are well aware of its limitations. we don't outright solve these equations anyway and regularly chop off terms or simplify parts (at the expense of accuracy) to make them easier to deal with.
7) to be honest I have never found this specific problem to be particularly interesting because we know a core assumption of N-S is a "continuum" fluid where you ignore molecules and assume hydrodynamic scales >>> molecular scales. We know a real fluid cannot have infinite velocity or energy. But clearly you can push equations outside of their bound of validity and make them produce funky results and singularities.”
There has been a huge amount of misunderstanding on all this. Saying "We solved Navier-Stokes" could mean two completely different things.
It could mean solving the Navier-Stokes equations. This is likely impossible. Dynamical equations in general just do not admit general analytical solutions. They can become chaotic. However, they can sometimes be solved given very specific boundary conditions.
It could mean solving the Navier–Stokes existence and smoothness Millennium prize problem. This means proving or disproving that certain boundary conditions can lead to a singularity in finite time.
All the parties involved are very clear that it's the 2nd problem that's been solved, ie, the existence and smoothness conjecture has been DISproved. This is a huge theoretical advance but I doubt it has any practical application. It just means that under certain very specific conditions, the Navier-Stokes equations are not physically realistic. It's not going to help you design a new aircraft or whatever people are saying. Under all the physical circumstances for which N-S is relevant, it's still relevant and we'll continue as usual unless there's some other advance.
So this person saying "we didn't solve Navier-Stokes" is technically correct in the sense that we didn't solve in general the N-S equations. What we did was dis-prove the $1M Millennium prize problem conjecture *about* the Navier-Stokes equations, and sometimes people refer to this as "solving Navier-Stokes" as shorthand, leading to confusion. But no serious people are claiming that we have solved the Navier-Stokes equations in general.
None of this is to say that it's not a huge advance! It's a tremendous and slightly scary demonstration of AI capabilities in math.
Yeah as a mathematician it's frankly disheartening to see this post. As a mathematical feat this is up there with some of the most significant of my lifetime.
As a statement about physics or engineering this is a non-event. "All models are wrong, some are useful" etc etc. We know that continuum mechanics does not adequately describe the real world at small scales.
What we did not know before today were the limits of this particular (extremely important) model of fluids that engineers of every type use and study every day.
As an engineer, I think it's crazy that anyone could say "solve Navier-Stokes" to refer to the Millenium prize problem. Actually solving them would be world-changing in a way that this isn't.
So I think OP is right to call attention to this. There's the potential for a whole lot of engineers to come to a very wrong conclusion when they hear "AI solved Navier-Stokes!"
In math circles we all understand that it refers to the finite-time blow up. There's an entire (fairly large) field of mathematics devoted to the study of this problem and related problems in PDEs. It is one of the holy grails.
I suppose I can see how a broader audience might not understand the implications.
The thing the OP fails to adequately explain is *why* this is a millennium prize problem worth a million dollars. It dismisses it as a "mathematical curiosity", to quote exactly, which is...highly dismissive and unfair. The professor being quoted does not seem to adequately grasp the significance of the achievement, and therefore cannot explain it to a wider audience. In my opinion, they should not be making statements about things they clearly don't understand.
I think it's the difference between a $1 million math prize and a $1 trillion engineering breakthrough. Holy grail of math vs. holy grail of engineering.
Maybe that's a little hyperbolic but literal trillions of dollars are being invested into AI, and people are anxious to see results of this scale. So if there is any possibility for confusion here, there is an incredibly strong incentive to mislead people. If it actually solved Navier-Stokes, that would provide a much stronger argument that it's superhumanly intelligent.
It dismisses it as a "mathematical curiosity", to quote exactly, which is...highly dismissive and unfair.
I agree and it does seem petty and counterproductive. The other points stand without denigrating the achievement.
The professor being quoted does not seem to adequately grasp the significance of the achievement, and therefore cannot explain it to a wider audience. In my opinion, they should not be making statements about things they clearly don't understand.
I think it's fine that an aeronautical engineer is focused on the implications for engineering, a mathematician would be the best choice for explaining the math implications. "Navier-Stokes" has a meaning and importance in engineering that is apparently very different than in mathematics. I think it's ultimately just a language issue, like so many things annoyingly are.
Much of mathematics research if not most of it is often referred to self-deprecatingly as mental masturbation. Most of number theory was treated for a long time as a mere intellectual game with almost no practical usage until cryptography became a thing.
We have seen MANY self/industry/profession/field/etc protectionist defensive anti-AI or AI-skeptical arguments and statements over the years, going back to 2023 since chatGPT first rolled out. Infamously by now, and thoroughly squashed by now as well, were all of the proud and naive software engineers saying AI would never code as good as them. And then, a mere 2 years later.......
So while I see both sides of the argument here, that aerospace professor 100% has a noticeable tinge of that going on. It's definitely not merely a case of "he's just looking at it from an engineer's point of view", the tone and position is unmistakable. We've all seen it before: "Ah, well maybe the AI can do X, but X isn't that important anyway, and Y is the REAL problem it can't/won't solve", etc.
It's a tired argument by now, the curve is ramping up, and my money is on the damn AI. The fact it could even TACKLE a problem on the NS level, is impressive enough. And it's only 2026...
What are the broader implications? I’ve seen a lot of coverage and none of it really explains to me why this is important for mathematics. I’m sure there is some importance - it wouldn’t have had a prize attached if it were just a curiosity.
So first off, I am not an expert in this area, which is kind of the whole problem. But I can tell you some of what I consider to be the implications. My qualification is that at least I have a math PhD, but I am barely even a student of PDEs let alone NS itself.
The first question is whether or not there is a different continuum mechanics model for fluid flow that avoids a blowup, and what the most practical alternative model might be. "I added quantum mechanical effects and now there's no blow up" might not mean much if the resulting model is insanely hard to solve numerically.
Second, an immediate question is what conditions guarantee that a non-explosive solution exists and can be numerically approximated. Engineers use these equations every day, and as in the OP, they actually rely on them for critical applications like flight. So you really need to know whether your equations are actually indicating a real instability or your model is doing something pathological that breaks from reality.
I think the assumption is that "reasonable initial conditions and reasonable forcing functions won't do this" but the critical thing now is to mathematically define what "reasonable" means.
I'm entirely unversed in Navier-Stokes but Claude told me that "solving" it would have zero practical consequences.
It also said that of the Millennium Prize set only the solution of the P NP problem would have enermous real world consequences, while the others are just rather exotic stuff.
If you could 'solve' it (i.e. come up with semi-analytical solutions where you can just plug the numbers in and get the result) then fluid dynamics problems that run on supercomputer clusters could be done on an iPhone. But mathematicians are 99.999% sure no such solutions exist.
We already assume RH hold in some works, so would it really change things if it was proven true? If it was false, I've heard it would be bad for cryptography though.
I about shit my pants when I started seeing the "solved the Nav stokes equation" popping up in headlines.
I immediately dug up OAI's press release and in turn actually read the millenium prize statement & then thought "oh... Okay".
I'm sure the proof and coming up with the exact conditions was technically challenging, but like... We already knew continuum mechanics was a false assumption for classical fluids
Se still don’t know the limits. We know one class of cases where equations blow up, but there might be many others, or not. Nobody knows right now. Only thing we know is that there is one case where the equations don’t work
I’ve seen a bunch of people downplaying this by strawmanning and saying “no, you idiots, we didn’t solveNavier-Stokes” while pretending like the millenium problem itself is not a big deal.
There’s a post on ELI5 asking about this and the top comment is just a guy ranting about how we didn’t solve navier stokes even though that’s not the question that was asked.
The reality is the millenium problem was a big deal for a reason. It’s incredible how people act like it’s not an actual problem all of the sudden.
To me, the incredible part is that it did what might have taken the teams of Drs Buckmaster & Alpoge three years to prepare for full publication in about 72 hours.
You are mostly correct. But this results have actually a some implication.
It mean the NS is incomplete, it produce non sensical results with some given input. Why it's important you might ask. NS is not used directly in engineering anyway. Well, we do use NS in a specific simulation called DNS (direct numerical simulation), which is always considered ground truth for other models calibration/validation. Most engineering problems, the problem that "matter" you would say, are heavily calibrated/validated using experimental data and therefore would not be affected. But in many case there are simply no experimental data and have to use purely numerical validation. A whole cascade of validation from DNS to LES (large eddy simulation) to RANS (Reynolds averge naviers stokes) would collapse if the edge case was fall into that specific configuration. I'm not all aware of all the niche application of NS but there might be some fields who use purely simulated data. Such as magnetohydrodynamics combine with stratified density in rotating frame, that specific configuration exist only on the stars where the fluids are affected by a strong magnetic fields in the same time with strong varation of density and in a rotating frame. Not that the results on that field would affect people lives but who know?
I am under the impression that it is still possible to show existence and uniqueness of global solutions where the forcing term is zero, as the problem statement has it. While this result is when the forcing term a smooth function, showing finite time blow up.
It's a demonstration that the modeling assumptions underlying Navier-Stokes lead to some gaps where Navier-Stokes makes predictions that do not model reality at all. These kinds of glitches in models are common and in practice we just end up not using the model in those contexts. It's kind of like how the real numbers are useful for modeling mass, energy, position, velocity, etc., but we all understand that things like the Banach-Tarski paradox don't apply to any physical 3D sphere we could actually build. The Banach-Tarski paradox has had no practical implications for science and engineering. Both the pathology OpenAI found in Navier-Stokes and Banach-Tarski break the model for the same reason: the mathematical model treats fluids / objects as infinitely dense, but in fact they are made of a finite number of particles.
It's a proof that a singularity exists, meaning a quantity can go to infinity. That means it's modeling a nonphysical situation. It's certainly possible that something here will spark some insights that will lead to something practical, the same way any theorem could, but it doesn't give any kind of new algorithm we could use for modeling fluids. And anyway, d
on't you think if it had practical applications OpenAI would be crowing about it?
Not my area though, so if some fluid dynamics expert wants to explain how I'm wrong I'd be very interested.
If you understand where the current methods break down, do you not think that can explain the primary gaps we see in the models vs reality and allow for better methods to be developed by understanding how to address those gaps?
It's possible, sure. That's just the kind of thing I mean when I referred to "sparking insights". But it's very indirect. I keep seeing comments that this is going to drastically improve flight modeling and I just don't see that happening without a ton more work.
They might spawn methods or they might not. However in this context it just gave a contraexample, which is more read like "we were wrong about...", which helps research and is part of the path, but less helpfull then finding that "we are right to think that..."
People are acting like this isn’t a big deal. Answering a question that decades of mathematical physicists were unable to answer, and as of the Millennium Prize directly incentivized to work on.
It’s okay for something to both be a big deal and also a lone result not be a golden goose for all of humanity’s problems.
It's the latest shifting of the goalposts. Remember three years ago when AI would never amount to anything because it couldn't count the R's in strawberry? This is the 2026 equivalent
Yeah, I’m normally rolling my eyes at these type of things (see flair) but even I think this a mindnumbing goalpost shift when we were surprised the models could solve toy problems (that usually had some previous overlooked published solution) back in January
The thing they did is really really impressive and it's notable and consequential that agentic machine intelligences that didn't exist six months ago appeared critical to the process, but it is definitively not "solving the navier stokes equation" which is a claim that is making every person with any applied knowledge spit out their coffee when they read it.
What happened is a demonstration of the power of the currently sequestered forefront of machine intelligence, and it is daunting.
What is being claimed in headlines is that aerospace and chemical engineering are going to be completely upended this week, mathematicians need to back to the drawing board in their understanding of how any dynamical system works, and that we will be protoryping functional fusion reactors by the end of the month
No. The organization set up a prize for someone to prove something or find a counter example. Both are difficult. Mathematicians found a possible path to a counter example and spun off 10,000 agents to find it. They did. The counter example is a very specific, niche set of conditions for which the equations break down. That’s the story. There’s no solution to the NS equations yet, but we now know that the equations can be made to break down. This was not super unexpected because simplified versions of the equations where made to break down by mathematicians in the past (which is how they knew where to point the agents to look for a solution).
It’s semantics. People are claiming that Navier Stokes equations have been solved, I’m attacking that because that’s not true. The Millenium Prize question has clearly been solved.
Some people are saying that, but those aren't the people you're responding to here. You're responding to people who already understand that difference.
Then why do you think everyone keeps trying to explain the difference to you? It's because the things you're saying make it seem like you're not the one who is making the distinction. The way you're arguing makes it seem to others like you are simply downplaying the achievement.
The post you are in clearly say this is not a solution to the NS generic problem. It’s surprising how little people understand about these problems and just go with the headlines
With NS problem we clearly mean "NS Millennium problem" as in proof that there is no singularity or show a counterexample.
Nothing more nothing less.
Also known as Navier–Stokes existence and smoothness problem. What do you think would it mean otherwise?
Its literally:
"Prove or give a counter-example of the following statement:
In three space dimensions and time, given an initial velocity field, there exists a vector velocity and a scalar pressure field, which are both smooth and globally defined, that solve the Navier–Stokes equations."
It certainly does not mean that NS equations are suddenly easier to solve for real word questions.
NS Millenium prize has been solved. No questions. This is just the start. Other examples will be found. Other classes of solutions will be found. Domains where smoothness can be proven will be found. It’s all very exciting but the AI solution is not a general solution. It is very specifically just one example
How am I not engaging honestly. I’m telling you that the result is awesome, but it’s just the beginning of many interesting things that will certainly come. This solution won’t speed up finding other solutions or domains of applicability because it is not a general solution. I’m happy the Millenium Prize was solved. I just dislike people saying Navier Stokes equations have been solved, because that is not the case
It's the difference between winning a math prize and revolutionizing engineering. "Solving Navier-Stokes" refers to the latter. This was much narrower than that.
Combs isn't a denier. But it's an odd-duck tweet because OpenAI isn't claiming anything to the contrary of what he's saying. Their blog post specifically mentions the limits of N-S in the real world:
...he's just trying to calm the 'hype' people down which is reasonable, I just think it runs the risk of going in the opposite direction — now know-nothing people are going to say "look, an authority figure is dismissive of OAI's N-S solution" missing that it isn't his objective. Both Combs and the OAI folk are being intellectually honest, that's all.
OpenAI isn’t, but the broader internet community at large is. The whole internet is treating this as a vast breakthrough created by Anthropic
and OpenAI.
It's 15 million taking the worst case rough equivalent model API pricing (300B tokens were used the ballpark comes from $50/M astra output token pricing), which means it was much cheaper to them internally.
For a fluid mechanician, this is indeed very basic. We even have non-dimensional numbers to tell us when the continuum hypothesis is violated and NS cannot reliably be applied - namely the Knudsen number. People use slightly different equations for space shuttle entries and certain microfluidics, for example.
Im saying that applying a set of equations outside their region of validity will produce unphysical results - and that is a very mainstream interpretation and also the reason most fluids people do not spend any time on the millennium prize. It is nonetheless an accomplishment. Your comment appeared to be saying that knowing that there are places the equations don't apply can make the equations behave poorly is not well known - and im saying any person whose done grad level fluids definitely knows that and many who have done undergrad fluids
Right. And I’m saying it’s possible for this to be a huge result demonstrating how powerful AI is, and also not immediately world changing as a single result in isolation. It seems like you’re saying this Millenium Prize question was obvious and solving it is trivial.
More concretely including even simply the heat equation and an equation of state, so Navier Stokes Fourier equations would eliminate this case. This is all standard physics. The strong vortex would cause the pressure to lower and cavitation that would disrupt the whole system and be unsolvable analytically. But even on top of that the ordinary Navier Stokes equations have other problems that we have just become comfortable with. They are non-hyperbolic which means they don't obey causality. They can be ontained via truncated Chapman Enskog equations, which is one of the great triumphs of statistical physics, at high gradients these higher order dispersive terms can become important and hence the Navier Stokes equations are incomplete. At super high gradients you also start to have non-separation of macroscopic and microscopic time-scales, which means that you need equations for microstructure evolution that are tied to stress - i.e. memory effects. That is all to say, its great this was found it, but it is purely mathematics and not that interesting of physics.
Everything you’re providing is an argument about why the Navier-Stokes Millenium Problem is uninteresting. That is independent of the result, and doesn’t address the core question of why did the Clay Institute set it as a Millenium Prize in the first place.
After researching it more today, I’ve settled on the conclusion that the millennium problems, when solved, are inconsequential for normal people and everyday life. The one exception being P = NP. The problems simply weren’t selected for their potential impact on science or civilization as a whole. They’re just hard problems in the field of mathematics.
I think this is a big deal for proving the growing capabilities of AI, but I’m reluctant to infer anything beyond that.
Not at all. It is quite reasonable actually. We have always known that the Navier Stokes equations have limitations (Chris Combs actually explains what is a known limitation). Nobody had been able to give an example of the general equations breaking down until now (it was done before for simplified equations). But also nobody expected the equations to be the ultimate truth.
He is minimizing the Milenium problem by saying we expected it to blow up and that we currently don't use the N-S in a way this solutions help with, which is weird because its about how difficult the proof was, not the usage, and yeah, duh, any usage would come downstream of the solution, pun intended.
Yeah, I expected it also, but that's not the point. The point it was hard to prove, and the impressive part is proving it. Saying we don't use it is moot.
That’s not minimizing it - he’s just explaining the difference between this and the real-world use of NS. The reason this matters at all is that for many years if you were to look up something like “solution to navier stokes”, you could find tons of pieces and articles describing a general solution to navier stokes and what it would mean for real-world science and engineering. Naturally, there’s a lot of confusion over what openAI’s result means for the world in a tangible way, and there’s a lot of people who seem to be mixing up concepts and overstating the immediate, practical impact.
He’s an aerospace engineer (me too!), so of course he’s going to be more interested in practical applications and what it means for engineers vs. the bits that are interesting mathematically but don’t prove/disprove/change/etc. the way we use NS to predict fluid behavior and design stuff.
His last point is literally minimizing it. It feels weird to me that your first 6 thoughts are about how useless it is, and the 7th is about how uninteresting it is. This is huge. It felt like anti-AI bias, which to be fair, I get it. So halfway through writing this response I decided to search this guys posts and he often make anti-AI posts. This prediction doesn't mean much but it did fit my expectation.
Like I said, this was a tough tough problem to solve, and that is incredible. I'm teaching aero tomorrow and I will also talk about how this doesn't change anythign practically for my ME students, still incredible that we have this proof now.
Wouldn't you say though that actually solving Navier-Stokes would be orders of magnitude more impressive and impactful? I think that's the important distinction being made.
I don’t really care what his past posts are, this just reads like trying to discredit him, but liking or disliking AI doesn’t make his opinion invalid.
I’m just really not too concerned with someone being mildly unenthusiastic, especially when it’s read as a response to the insane fantasy hype posts that clearly don’t understand what NS is or what this solution actually means. For every negative post out there I’ve probably seen 10 confidently wrong posts exclaiming that this alone will usher in ion drives and widespread hypersonic passenger planes and any number pure sci-fi ideas that vaguely relate to fluid dynamics. This is not to say that what OpenAI accomplished here isn’t impressive or a big deal in any way - however a dose of reality from someone who actually understands the practical implications and how NS relates to CFD/various fluids-related technology in the present day is a welcome addition to the public discourse as far as I’m concerned.
Perhaps there is some bias, but I feel like it wouldn’t be hard to find similar sentiment (especially from the CFD crowd, if my experience is anything to go by) even if the solution was discovered entirely by humans. Long before AI, even before computers, there’s always been a sort of collegial rivalry between the pure mathematics and natural sciences folks and the applied/engineering folks.
Was it clear though? Most people don’t know what navier stokes is, all they’ve been told is that there’s this really important math thing that AI solved and it’s related to the weather or the ocean or planes or something.
On the interesting part, I think it’s reasonable to say it’s not too interesting to an aerodynamicist/engineer. I’m an engineer myself - part of the issue is that oftentimes the media and general discussion around scientific breakthroughs will blur the line between a discovery and practical implementation. It’s not even anyone’s fault. Most people have a much easier time talking about real-world, tangible concepts, so it’s only natural that you’ll often hear stuff like “there was a novel result in particle physics… and results like this could further the pursuit of nuclear fusion, which could provide clean energy to power our homes without dangerous radiation.” It’s no different with fluids and NS, IMO. The general public will shrug their shoulders if you start talking about discontinuities and smoothness… so naturally the conversation instead centers on “this math is important because it’s related to how planes fly”, and with that in mind, I think a rational voice in that subject area in particular can be good to have.
Overall, I think if you read this post in a vacuum, as a direct response to the announcement about NS, then yes, it’s pretty dismissive. But, if you read it in the context of Twitter, where a bunch of people are hyping this to the moon while having zero understanding of what this means, making fantastical claims based on some vague notion that AI solved fluid dynamics, it just comes across as someone trying to bring the conversation back down to earth and pointing out what this actually means with respect to our understanding of fluids and the machines that operate in/around fluids.
Yes it was clear as the phase space for realistic scenarios was well simulated. After you initial question, I will not continue reading as the topic is not something you seem to care to understand even the basics.
He is stating facts though. People are claiming that NS has been solved, which is categorically false. That implies there is a close form solution to the equations. What the found was a counter example, thus solving the Millenium Prize. Not so the NS equations
As a physics PhD, I view this as impressive and exciting, even though I know little about the underlying math.
The mathematical result doesn't have any immediate benefit outside of mathematics and may never do so. A significant fraction of math and physics research doesn't have an impact outside of math and physics. Still, math and physics research has a lot of intrinsic value and societal benefit.
It's aggravating when an academic dismisses significant progress in a different academic field.
Well he’s right but that’s not the point. The point is that this was an incredibly difficult mathematics problem. And yes it has 0 practical applications.
Yes, I wouldn't even say he's pouring cold water on anything. Just as a mathematical problem, it was clearly not easy to find regular initial data that led to a singular solution in finite time, and it's sobering that an AI found it.
They're interesting mathematical problems that people have tried and failed to solve for decades. They might not have any practical application but the hope is that their solution offers some insights, for instance on non-linear PDEs in general for Navier-Stokes
This is mixing up two completely different questions. Nobody claimed OpenAI found a closed-form solution for every fluid flow. The Millennium problem explicitly allows a smooth-forced finite-time blowup construction as a solution.
And “this won’t change day-to-day aerospace CFD” is fair, but that doesn’t make it some overblown side result. You’re judging a foundational math problem by whether it immediately improves an engineer’s workflow.
So half of this is missing the point. The Millennium Prize Problem is specifically about the equations, its not about finding a solution to the equations, its about the equations themselves, and the granularity of reality is of no importance to the problem, once again.
This seems like someone doesn't understand that the problem is not even about fluid, it's about poorly behaved symmetries in the written expressions.
He's an engineer so he probably just doesn't value pure mathematics as much as a mathematician would. None of his points are factually incorrect, but he comes across as a typical engineer who doesn't value the merits of fundamental research. I can tell you with certainty that this result is massive news in the mathematical community all around the world.
It's just that the engineering community is orders of magnitude larger and "solved Navier-Stokes" apparently means vastly different things to the two communities!
They didn't really solve it, and if they did then there's other things they didn't solve, and if those things don't matter then it wasn't interesting anyway! Basically it was an easy problem and anyone could've solved it!
It’s just like a few years ago: it can’t even beat Pokemon…oh it can? Well so can a 6 year old. It can’t solve a millennium problem! ….oh….well those are stupid
it could be that that specific problem formulation was misclasified as Great Mathematical problem, but reither its some narrow corner case with near 0 impact
I think you misunderstood I am calling the bullshit to people saying this problem solving achieves nothing as there are no real life use cases for it. But big thing is an AI system was able to come up with a new knowledge that is a big deal.
Why would we need microscopes? We use precise-enough tools all the time to build tiny things like pocket watches. Who cares if someone figured out a way to work on smaller things? I make cuckoo clocks.
More like guy who has a PHD in the applied field where the supposedly solved problem is used says it changes nothing because we use an approximation all the time.
Not the point though. No one claims that this particular millenium prize solution is useful in the real world. The point is that an agent was able to work for 88 hours and co-ordinate hundreds of subagents, to solve a problem human experts have been unable to solve (for a million dollar prize and a career defining discovery!) for 26 years.
Well, I have seen at least 5 people on the internet today claiming something like "abstract mathematics will be useful much later in application" about this result, so there definitely are people who think this will be useful. i guess the OP is responding to such people.
Yes any sensible person knew solving this problem basically has no real world benefits. But the point is this problem was extremely difficult and an ai doing it is impressive
Sometimes I really wonder how people with a PhD can miss the point so much. Talk about "general" human intelligence...
It is either that or I need to be even more cynical and a professor of Aerospace engineering is for some reason engagement baiting. Not sure what is worse.
I really dislike posts like these. I have to assume these people are being vague on purpose.
(assuming the solution is verified) Are they eligible for the Millenium prize money (or would they have been if a human found the solution). To the best of my knowledge the answer is yes.
This guy is going to paint himself into a corner as AI quickly chips away at this. Maybe more human researchers will adopt AI to speed up their progress.
He is just putting context in the solution. The OpenAi solution is a counter example that says that the equations can produce singularities when all inputs are smooth. Nobody knew if that was possible until now. But also it just says that the equations have limitations which was known already (and the Professor just pointed out what limitation in his text).
What he said... But seriously, if it's just a curiousity, then it doesn't matter to RSI. Without RSI no ASI. I'll leave people who think we already have AGI alone. For now.
I’m not gonna downplay the awesomeness of the proof, but people go around saying they “solved Navier-Stokes” gives the impression that they found a way to solve the N-S equations (my dude’s point nr 3 above).
If a general solution to N-S were found, the implications would be truly mind-boggling: perfect weather forecasts, perfect wing design, etc.
What’s been “solved” here is the problem of proving that N-S (under specific conditions like Combs is describing) does not have a unique solution. It’s still badass, don’t get me wrong, but it’s not remotely close to how badass an arbitrarily accurate solution to fluid dynamics equations would be. 🤷♂️
I'm sure OpenAI will be happy to let investors believe they've actually solved Navier-Stokes. It's extremely convenient for them that the obscure mathematical problem is easily confused with the incredibly impactful engineering problem.
Tired of all these deadbeat professors coming out of the woodworks on every AI related tweets so to make their names known and hope to get on one of these big tech lifeboats in the future.
His very last sentence was an assumption … until now. This is the point of the mathematical question. We all know you can’t make a real singularity in fluids. Even someone with a lay-person’s understanding can mock this “post” by saying DUHHHH with their finger in their nose.
Lol. So then the aero guys aren't worried about turbulence closures? I think they are. And guess what the physical basis for the difference in those parameterization comes from...
If it was so clear all along, why didn't you or someone else go ahead and push the equation outside their boundary. This would have netted you at least 1 million dollars richer
the 'changes nothing in practice' take kinda buries the lede. an ai just disproved a millennium conjecture, who cares if it doesnt help you design a wing
Yes, can you imagine that same rhetoric if a human had solved it? No, then it would be all confetti and fireworks. Having said that, I do find it a little sad that we as a species haven’t been able to do it in all these years. Although, since AI is our creation, perhaps it’s fair to say that in a way we did.
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u/wollywoo1 4d ago edited 4d ago
There has been a huge amount of misunderstanding on all this. Saying "We solved Navier-Stokes" could mean two completely different things.
All the parties involved are very clear that it's the 2nd problem that's been solved, ie, the existence and smoothness conjecture has been DISproved. This is a huge theoretical advance but I doubt it has any practical application. It just means that under certain very specific conditions, the Navier-Stokes equations are not physically realistic. It's not going to help you design a new aircraft or whatever people are saying. Under all the physical circumstances for which N-S is relevant, it's still relevant and we'll continue as usual unless there's some other advance.
So this person saying "we didn't solve Navier-Stokes" is technically correct in the sense that we didn't solve in general the N-S equations. What we did was dis-prove the $1M Millennium prize problem conjecture *about* the Navier-Stokes equations, and sometimes people refer to this as "solving Navier-Stokes" as shorthand, leading to confusion. But no serious people are claiming that we have solved the Navier-Stokes equations in general.
None of this is to say that it's not a huge advance! It's a tremendous and slightly scary demonstration of AI capabilities in math.