On Monday, OpenAI announced that a swarm of its AI agents had resolved one of the Millennium Grand Challenge problems, the Navier-Stokes problem. The Millennium problems are seven landmark mathematics questions posed by the Clay Mathematics Institute in 2000. A successful solution confers immortality in the pantheon of mathematics heroes plus a million dollar prize. Before OpenAI’s announcement, only one of the problems, the Poincare conjecture, had been resolved by the reclusive Russian mathematician Grigori Perelman. He refused the prize, rejected the fame, and disappeared from mathematics.
You would think the mathematics community would be rejoicing but OpenAI’s announcement has been met with acrimony and cynicism. Two human mathematicians, NYU’s Tristan Buckmaster, and Anthropic’s Levent Alpoge, were zeroing in on the problem and were well on their way there. But when OpenAI heard that Anthropic was working towards a solution to the Navier-Stokes problem, they threw 10,000 agents at the question and 105 hours later had a 166 page proof.
Buckmaster subsequently claimed that openAI offered to share the credit with him, but only if he omitted the contributions of the Anthropic researcher Alpoge. Buckmaster also raised the question of whether OpenAI had used his and Alpoge’s research or chats with OpenAI models in its training or model runs. OpenAI has denied these accusations.
You might be tempted to dismiss this controversy as nothing more than academics fighting about who gets academic credit. But it’s much more important than that. We are seeing perhaps the first real skirmish in the conflict between the primacy of human intelligence and artificial intelligence. The foundation labs want to make artificial intelligence primary, even though they constantly kvetch about how dangerous this is. OpenAI wants to give credit to its models for solving the Navier-Stokes problem. Human mathematicians are balking. They want to give human mathematicians the credit.
Of course, if you are just looking at the headlines, it would appear that a super-intelligent AI solved a problem that has long eluded humans. That’s not really true though. To see why, it will be necessary to get into some detail on what actually happened here. Human mathematicians have a good case that the people who did the essential work should get the credit for the Navier-Stokes problem.
What are the Navier-Stokes Equations?
The Navier-Stokes equations govern all fluid flow—the flow of water in a pipe, the flow of air around the wing of a jet, the flow of blood around a stent in the heart, the flow of air and water that creates the earth’s climate, and even the flow of plasma in the sun. Given some initial conditions and some forces applied to the fluid, the Navier-Stokes equations allow the prediction of what will happen to the fluid at each point in space and time.
The equations may look intimidating to the uninitiated, but they just encode two important relationships from basic physics. The first equation says that mass is conserved. That means that the flow of a fluid into a region must equal the flow of fluid leaving the region. Fluid—mass—is not being created or destroyed.
The second Navier-Stokes equation is a complicated version of Newton’s second law of physics. As you may remember, the second law says
Force equals mass times acceleration.
If you have a mass sitting at rest or moving at a uniform velocity, the only way to accelerate the mass, i.e., change its velocity, is to apply a force to it. In the second Navier-Stokes equation, the role of mass is played by the fluid’s density, which is mass per unit of volume.
In the case of fluids, Newton’s second law says that density times acceleration is equal to the sum of three kinds of forces that apply to fluids. The first force is the force applied when a fluid moves from a region of high pressure to a region of lower pressure. The second force arises implicitly from the viscosity of the fluid. Viscosity is like friction for fluids. When fluids rub up against each other, they apply forces to each other. The third force is whatever external force happens to be around affecting the fluid. For example, gravity is a force that affects fluids. On the sun, the external forces that affect the flow of plasma are gravity as well as the electromagnetic forces, among others.
Why are the Navier-Stokes Equations a Millennium Problem?
Conceptually the Navier-Stokes equations are simple. So, what is it about them that lands them on the Millennium problem list? The difficulty is not in the density term nor is it in the three forces per se. The difficulty is in the acceleration term, which as you may recall, is the rate of change of velocity with time. When you are talking about fluids, acceleration is complicated: it has a non-linear component. The non-linear component arises because velocity depends on the time and on the position of a point in the fluid.
For example, if a fluid is flowing through a pipe, the velocity of the fluid can vary with the diameter of the pipe. So, the velocity of a point in the fluid depends on exactly where it is in the pipe as well as on the time you look at it. The dependance of velocity on time and position introduces a feedback loop. It’s that feedback loop that induces the non-linearity.
Non-linearity is a problem for physical equations. If the Navier-Stokes equations were linear, you could be confident that if you made small changes to the initial conditions of the fluid—such as the initial pressure and the geometry of what the fluid is flowing around—and you also made small changes to the external forces, you would get proportionately small changes to the predicted velocity of the fluid at every point in time and space. But if the equations are non-linear, you can’t be sure what will happen. It may be that small changes always produce small velocity changes. Or it might be that small changes in the initial conditions and the external forces produce changes in the fluid that drive the fluid velocity to infinity. Mathematicians want to know if there are situations in which small changes to the initial conditions and to the forces can make the fluid velocity increase to infinity while not violating any physical principles, such as conservation of mass or energy.
How Was the Navier Stokes Problem Resolved?
For over a century, mathematicians didn’t know which situation applied. The Millennium problem could be solved by proving that either
the velocity of the fluid always stays finite for any smooth initial conditions and force, or
there is a counterexample in which the fluid velocity can increase to infinity even though the initial conditions are smooth, the forces applied are finite and smooth, and no physical laws are violated
(Notice the requirement that the force be “smooth.” That condition will turn out to be critical.)
OpenAI went with option 2. OpenAI’s agent swarm resolved the Navier-Stokes problem by constructing a counterexample and showing that it was a legitimate proof using machine verification. Of course, the counterexample will have to be verified by human mathematicians before it can be accepted as a final resolution of the problem.
The counterexample is striking. When finite external forces are carefully applied just so, the fluid will whirl in a vortex, going faster and faster, while the vortex shrinks, becoming thinner and thinner. The volume of the vortex collapses fast enough that the kinetic energy remains finite as the speed of the vortex goes to infinity. OpenAI’s paper depicted the vortex as follows:
What was the Role of Human Mathematicians?
The OpenAI solution was possible because human mathematicians had already invented a new strategy for attacking Navier-Stokes style problems. Previously, mathematicians had tried to come up with a proof that velocity could blow up to infinity in some complex fluid flow. That strategy never worked. Spanish mathematicians Diego Cordoba and Luis Martinez-Zoroa tried a different direction in the early 2020s. They observed that you could specify some fluid velocity and then use the Navier-Stokes equations to solve for an external force that would have produced it. Perhaps you could construct the infinite velocity example and then use the Navier-Stokes equations to derive the force that produced it. That strategy might produce the counterexample required by option 2.
Their idea was to cleverly add components of velocity up so that the velocity would go to infinity, but when you solved the Navier-Stokes equations for the external force that would have produced that infinite velocity, the force would remain finite. To accomplish that seemingly impossible task, they had to add the velocity terms up in a very clever way that exploited the non-linearity of the Navier-Stokes equations so that enough quantities cancelled out to keep the derived external force finite. In 2023, they demonstrated their strategy worked for the Euler equation, which is a special case of Navier-Stokes with zero viscosity.
Buckmaster and Alpoge then adopted the Cordoba/Martinez-Zoroa method to see if they could extend it to find a counterexample to the Navier-Stokes problem itself. The Spanish mathematicians’ method couldn’t be used as is because it yielded a force that wasn’t “smooth.” Smoothness of the force is a requirement to solve the Navier-Stokes problem. If a force is smooth, it means the rate of change of the force with respect to position or time never changes abruptly. It always changes smoothly. But smoothness also implies that the rate of change of the rate of change doesn’t change abruptly. And the rate of change of the rate of change of the rate of change doesn’t change abruptly. And so on ad infinitum.
Smoothness is an essential condition because you could always cheat by getting an infinite velocity from a force that’s pathological in some way. You want to rule out cases in which there is something mathematically weird going on with the force.
Buckmaster and Alpoge’s essential contribution was to arrange the cascading of velocities even more cleverly so that the velocity becomes infinite but the force that caused it was finite and smooth. With substantial help from chatGPT and Claude, they showed that the method worked for the Euler equation, the Navier-Stokes special case with zero viscosity.
Buckmaster and Alpoge would have then turned to the full Navier-Stokes model, but they didn’t get the chance. Hearing that Anthropic was working on the problem, OpenAI hurled the full might of its latest models at the problem. OpenAI’s swarm of agents, building on the human work, did the further extension of the Cordoba/Martinez-Zoroa method to the case in which the viscosity term is left in—the Navier-Stokes equations—while keeping the force smooth, thus satisfying the requirements to solve the Millennium Problem. This is not a trivial extension. It’s an important advance. But, as we’ll see, the extension could have been made by Buckmaster and Alpoge without the swarm’s help.
What Did OpenAI Really Add?
Terence Tao, an authority on the Navier-Stokes problem, examined Buckmaster’s and Alpoge’s methods and praised their work as seminal.
Tao concluded that there is no reason that Buckmaster and Alpoge’s methods could not have been extended to solve the Navier-Stokes problem. Tao even speculated that that extension work could be delegated to AI agents, since he didn’t believe the extension was the most important part of solving the Navier-Stokes problem.
But then OpenAI came along and did just that, employing a swarm of agents to solve the full problem, thereby denying Buckmaster, Alpoge, and Cardoba/Martinez-Zoroa eternal mathematical glory. And they did it to avoid being scooped by Anthropic.
Ironically, Anthropic wasn’t even working on the problem. Alpoge, although he is employed by Anthropic, was working on the Navier-Stokes problem on his own time.
The Fallout
The American Mathematical Society (AMS) credited the human mathematicians, while not mentioning the AI agents. They referred to the OpenAI and Claude tools that Buckmaster and Alpoge used as “new technologies” and emphasized that the purpose of mathematics is human understanding, not machine glory.
Tao went further, questioning whether the affair would lead mathematicians to keep their research secret to avoid being front run by the giant AI companies. Scientists in other disciplines are probably asking themselves the same question right now.
Will Man or Machine Ultimately Get the Credit?
Eventually, the Clay Mathematics Institute will have to decide who gets the credit, along with the $1 million dollar prize. The AMS has signaled its view: it should be the mathematicians, not their tools, who get the credit for resolving the Navier-Stokes problem. The AMS is right. AI models are tools we’ve invented to make us more productive—nothing more.
We’re likely to see this controversy coming up again and again as the major labs claim that machine intelligence solved this intractable problem or that one, discovered a new physical theory, or made some major progress in biology. In the history of science and math, we often see important discoveries made around the same time by different people or teams. The invention of calculus by Newton and Leibniz is a famous example, but there are countless others. Relatively simultaneous discovery happens because most of the pre-requisite work has already been done and the conditions are ripe. Now, when relatively simultaneous discoveries are made, some will claim the machines did it.
Going forward, humans working with machines will be making the discoveries. When the conditions are right to make a discovery, anyone with overwhelming AI resources, whether foundation lab or government, will be tempted to jump in and take the final steps, claiming priority. That’s bad for open science, since scientists will start concealing their work. We should punish that behavior by always assigning credit to the people, not to their tools. The Clay Institute should start by giving the prize for Navier-Stokes to the people who took the decisive steps.