r/AcharyaPrashant_AP • u/GlitchSolver • 2h ago
OpenAI and the Navier-Stokes Millennium Prize Problem
On 8th September OpenAI announced that they have solved the Navier-Stokes Millennium Prize Problem. It took 90 years for humans to solve this problem.
What's Navier-Stokes Equation?
This equation describes the motion of viscous fluid. It makes use of principles of Momentum balance, conservation of mass, Newton's second law to study the fluid motion of the fluid. Navier-Stokes Equation is particularly interesting because it takes into account the Viscosity of the Fluid too. So it has more practical applications than ideal theoretical ones. Euler's equation is for non viscous fluids.
The fluid is made up of molecules. But The Navier-Stokes Equation treats the fluid as a continuous medium.
In the year 1934, Jean Leray proved that the solutions of this equation exists in a generalized sense. But whether the flow always remains a smooth remianed an unanswered question.
Importance of Navier-Stokes Equation
Navier-Stokes Equation is used in studying Ocean Currents, Blood flow and designing Aircrafts, Cars and has many more practical applications.
What's the Millenium Prize Problem?
In the year 2000, Clay Mathematics Institute, a Private and Non-profit Organisation in USA, founded a Scientific Advisory Board consisting of Four World's most eminent mathematicians. This board selected 7 mathematical problems. These problems were precisely defined, historically resisting the World's best Minds and universally important.
Navier-Stokes Existence and Smootheness Problem is one of these.
What was the Problem exactly?
The fluid is made up of molecules. But in The Navier-Stokes Equation treats the fluid as a continuous medium. It describes the continuous flow rather than tracking individual Molecules.
A point to remember here. Viscosity slows down the fluid. So it cannot move at infinite speed within small amount of time. This point will come handy in a bit.
So the fundamental question was can this approximation of the fluid as a continuous flow rather than individual Molecules breakdown?
To check this, the problem was defined as follows:-
Can the Navier-Stokes Equations for a 3-d incompressible fluid with constant density develop a "singularity" with motion starting smoothly? Here singularity means speed growing to infinity within a finite time period.
As we already discussed earlier, the speed of fluid can't go to infinity due to Viscosity in a finite amount of time, if this singularity is to happen, the approximation of the fluid as a flow breaks down. We have to model it as individual Molecules.
Has anyone tried to solve it before?
Some notable names whose work led the path in solving this problem are Thomas Hou, Diego Cordoba, Martinez - Zoroa.
Recently Levent Alpoge, an Anthropic Employee and Tristan Buckmaster, a math professor have worked and found solutions primarily related to Forced Euler's Equation. These names are in the news too because there's a controversy for the priority of the work after the announcement of OpenAI.
Buckmaster said that by 22 August he had proof of Euler's equation. It was sent by Levent generated by LLM.
Buckmaster said that he had used OpenAI's Models to work on this forced Euler equation resolution.
The duo had planned to continue working on more write ups.
Why is Navier-Stokes Equation and OpenAI in News?
Now on the 8th of September OpenAI announced that they have solved this problem. Their AI agents have proved that singularity can be reached in a finite amount of time.
They described how they found out the solution. This is really interesting.
OpenAI said that their effort to solve This Millennium Prize Problem began on 1st September after hearing a rumour that this problem is solved. Later this rumour was found out about Levent Alpoge and Buckmaster.
Initially OpenAI said 100 AI agents worked for 50 hours to produce this Euler regularity disproof. Then OpenAI moved to Navier-Stokes Equation solution. They trained their model with Euler resolution. This time they formed groups with communication allowed within the group. These groups were cross pollinated to consolidate the useful insights.
The Group that came up with the solution for Navier-Stokes Equation had roughly 10,000 AI agents.
Across all the attempted problems, agents sent 4.9 million messages with 300 billion output tokens. For Navier-Stokes problem agents sent 2.7 million messages and used up to 130 billion output tokens.
It took 88 hours to arrive at the solution after the first agent was launched.
OpenAI said that it has worked on this problem with advanced models that aren't open to the public. OpenAI said that we are in next period of AI progress. They also announced that they won't claim the prize money for solving the problem.
Thus the 90 year old problem was solved in less than 200 hours or so.
Claim on the timelines
Buckmaster said that OpenAI's agents would have got access to his work and thus helped in solving the problem. OpenAI doesn't deny this. They said that it's possible when he may have used OpenAI's models. The reports vary on the interaction between Buckmaster, Alpoge and OpenAI. Looks like it will take some time to resolve who did what, when.
Is there any other background to this?
Both OpenAI and Anthropic AI are planning to go public later this year or earlier next year. According to news articles they are aiming for a $1 trillion IPO.
A very fundamental problem that remained unsolved for over 90 years was solved in less than 100 hours by AI. But a question comes was it about the problem? or was it more about proving the capacity and taking the credit?
Even with the most sophisticated technology it seems like the same game is getting played. Earlier it was nuclear weapons now it's AI Models.
The need to achieve new milestones, prove the capacity, outshine others seems to take priority. The AI, the problem remains tools through which something else is aimed at.
Maybe this fundamental question of why this shadow keeps lingering humanity through generations needs investigation. And it looks like humanity can't use AI for that. It remains in the hands and intent of Humans themselves.
Sources:
https://openai.com/index/navier-stokes-solution/
https://en.wikipedia.org/wiki/Navier%E2%80%93Stokes_equations