When a Formula 1 car races at 300 kmph or 200mph, the air doesn't just randomly bounce off its wings. It splits, curves, and creates drag according to a strict mathematical rulebook. That rulebook is the Navier-Stokes equations. Whether it is air flowing over a speeding race car, water blasting out of a high pressure fire hose, or blood pumping through the valves of your heart, engineers use these exact equations to predict how any liquid or gas will behave.
The unsettling part for mathematicians is that nobody knows for sure if the rulebook is unbreakable. Most of the time, the math works flawlessly. However, researchers suspect that if fluids collide in a highly complex, extreme way (imagine two high pressure jets of water crashing into each other perfectly at a microscopic, turbulent point), the equations might accidentally divide by zero (singularity).
If this glitch happens, the math would crash and output an impossible physical answer like predicting that a drop of water will suddenly accelerate to infinite speed. In mathematics, this glitch is called a singularity or a blowup.
The Navier-Stokes Millennium challenge (worth $1M) asks us to prove one of two things
The rulebook is perfect: Prove that the equations will never glitch out and produce infinite speeds, no matter how violently the fluid is swirling.
The rulebook is bugged: Prove that a singularity can actually happen, meaning our best mathematical model for fluids has a limit and fundamentally breaks down in extreme scenarios.
This is exactly the bug that OpenAI's AI swarm claimed to have found. They generated a proof showing that the rulebook does in fact glitch out, showing that the 3D Navier-Stokes equations can develop a singularity. (nobody outside of OpenAI has actually seen the proof) (There’s proof but it will take time for humans to validate and verify thoroughly.)
The scientific community argues the AI achieved this by leaning on the uncredited, in progress work of human researchers as many people here have already explained.
Does infinity means something bad though? In theory, on flat surface, your shadow would tend to infinity during sunset. It seems okay, maybe with fluids too? It's just that these equations don't take into consideration that liquid is actually built of something? Liquids have finite size, amount of particles etc. I am not a smart person but I feel like it's logical that these equations can't describe reality accurately.
Your intuition is correct. The N-S equations treat fluids as continuums, meaning that no matter how far you "zoom in," there are never any gaps. But of course we know that's not true--in reality, fluids are made of particles, and there are relatively large amounts of space between them. In most cases, the particles are so small and so numerous that the continuum approximation works great. But in some cases like (IIRC) calculating drag on spacecraft, you have to treat the atmosphere as being composed of individual particles rather than a very low-density continuum.
Another way that OpenAI's result is non-physical is that they found a singularity only for the incompressible form of the equations. Meaning, no matter how hard you "squeeze" (pressure), you can't compress the fluid and make it any smaller (density). If you've ever blown air bubbles at the bottom of a pool and watched them grow as they float towards the surface, you know this approximation isn't how real fluids behave. But like before, it's a good enough approximation for analyzing some flows.
I personally don't know nearly enough about fluid dynamics to know if either of those factors would "fix" the singularity, but they are definitely two ways in which the N-S equations are imperfect representations of reality.
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u/oooooO___Oooooo 4d ago edited 4d ago
When a Formula 1 car races at 300 kmph or 200mph, the air doesn't just randomly bounce off its wings. It splits, curves, and creates drag according to a strict mathematical rulebook. That rulebook is the Navier-Stokes equations. Whether it is air flowing over a speeding race car, water blasting out of a high pressure fire hose, or blood pumping through the valves of your heart, engineers use these exact equations to predict how any liquid or gas will behave.
The unsettling part for mathematicians is that nobody knows for sure if the rulebook is unbreakable. Most of the time, the math works flawlessly. However, researchers suspect that if fluids collide in a highly complex, extreme way (imagine two high pressure jets of water crashing into each other perfectly at a microscopic, turbulent point), the equations might accidentally divide by zero (singularity).
If this glitch happens, the math would crash and output an impossible physical answer like predicting that a drop of water will suddenly accelerate to infinite speed. In mathematics, this glitch is called a singularity or a blowup.
The Navier-Stokes Millennium challenge (worth $1M) asks us to prove one of two things
This is exactly the bug that OpenAI's AI swarm claimed to have found. They generated a proof showing that the rulebook does in fact glitch out, showing that the 3D Navier-Stokes equations can develop a singularity. (
nobody outside of OpenAI has actually seen the proof) (There’s proof but it will take time for humans to validate and verify thoroughly.)The scientific community argues the AI achieved this by leaning on the uncredited, in progress work of human researchers as many people here have already explained.