I Never Thought I’d See This Happen

I did not think I would live to see the day when this mathematical problem gets solved. And I just did! Likely. So I am kind of sitting here in disbelief. Something that nobody has ever been able to prove, the Navier-Stokes existence and smoothness problem has been very likely solved, OpenAI says. This is history in the making. And there is going to be one even bigger surprise at the end. I will note that I have no relationship of any kind with OpenAI. They don't even send me things for early access. And they send out a lot of those. Not for me. Now, there is controversy. I am a research scientist, and this is very uncomfortable for me. But I have to make note about it, because it is external context for the work that you should know about. Then, we talk about the part that almost nobody is talking about. So, 2 scientists had made meaningful progress on a related problem, and according to legendary mathematician Terence Tao, they got to the point where their solution could likely be extended to solve Navier-Stokes. And then, after hearing the rumors, OpenAI fired up an AI system that is more powerful than Astra that came up with a solution. Whoa! Now, this is Two Minute Papers, where we celebrate science and scientists, and I do not know what kind of attribution these 2 scientists outside OpenAI will be given, so as a thank you for their work, here is our attribution to them. And I also want to say a big thank you for their work. One more note. These scientists also used proprietary LLMs like ChatGPT and Claude extensively. And they were looking for verification that OpenAI did not reuse this data in their systems as training data or otherwise. And to this, here's the answer. One sentence. Officially from OpenAI. While unlikely, we cannot rule out that de-identified data derived from their usage of our products helped improve our models. Yes, whatever you enter in a chat box for a proprietary AI can be used for training. I keep saying this over and over again, that this cannot happen if you run free and open weights AI systems yourself. Why? Because the prompts never leave your machine. Okay, Navier-Stokes. Finally, I get to talk about science. I am not an expert on this, I am just a student. But I have a little experience in this area, as I did a few years of research and wrote my thesis in this area. Now, what are the Navier-Stokes equations? These are equations that describe fluid motion. But what? I mean, that sounds crazy! Look at this absolutely beautiful nature footage. I remember looking at this and saying, you want to understand and compute this? Are you crazy? This is unfathomably complex! Well, it turns out, with the power of science, it isn't. The Navier-Stokes equations teach us that you just need to understand 3 terms, and you understand all of this. First, advection. This is the heart and the bane of every fluid simulation. Here's the good news: if you drop an object into a river, it will follow the flow. Goodbye! This is advection. Bad news: the fluid also advects itself too. This is described with the directional derivative in the first term. Second ingredient: pressure. Yummy. It's a bit like people on the bus. Where there are a lot of people, there is a lot of pushing each other, which starts outwards movement. Third ingredient, diffusion. Crowd favorite. This means that differences average out over time. If you add a drop of ink into water, you immediately see where it is. Of course. However, if you come back in a few minutes, you see that it has spread perfectly, and the whole glass of liquid is now the same color. That is diffusion. Also, the liquid reacts to external forces. If you blow on it, it moves. Of course it does. Done with a beautiful, simple addition. I did not count this as a term. You also need to add a second equation, the incompressibility condition that says that volume remains constant over time. We don't lose or add liquid out of thin air. And what I love to do is to discretize this kind of equation onto a grid. And these expressions are very simple to evaluate on a grid, for instance, advection means that you take some fluid density out from where you are, and add it to the appropriate neighbor. Diffusion means averaging. So simple. With this, you can write a computer program that kind of simulates reality. That is mind-blowing. Fun liquid simulations? Yes please! Wind tunnel tests for a new aircraft? Yes please! Karol writing up crazy simulations to control the fluids of the world? Yes please! You can write a simulation like that too. My source code for all my research is always available free of charge for everyone. With the papers too, links in the video description. Okay, so what is the question to be solved here? Well, the question is, if you start with a smooth fluid flow and run these equations forever, does the mathematics eventually Dear Fellow Scholars, this is Two Minute Papers with Dr. Károly Zsolnai-Fehér. And I am happy to report to you Fellow Scholars that the answer is not 42, as some of you say. The million-dollar answer is— yes, the mathematics can break down. The equations are not guaranteed to behave nicely forever. OpenAI starts the solution from rest. and uses carefully created external forces. You see, if you create a vortex that spirals inward, stretches, and its velocity grows without bound within a finite amount of time, while its total amount of energy remains finite, then the mathematics breaks down. So, there are mathematical cases that are hairy. Does this happen in Mother Nature? Well, not that we know of. If it did, molecular-level physics would take over. Don't use Navier-Stokes for that. And once again, I am just a student who is trying to learn here, I may be wrong. But I am trying my best. Now, another surprise. How long did it take OpenAI's model to find out? Have they been running it for a year or longer in stealth? Nope. Now hold on to your papers, Fellow Scholars, because it took about 3.5 days. Goodness. And one more thing. People don't really understand why AI is so good at math. Let's look at prose. If you ask an AI to write an article in your style, so it does. What do you do? You read it and evaluate it by hand. Was it good, or was it not? Well, you read it and evaluate maybe 100 of those per hour. But with mathematics, not so much. Math is verifiable. That is the key. You can check if it's good or not automatically, so you can do it 100 million times per hour. 100 lessons versus 100 million lessons per hour. There is a big difference. This is why AI is improving incredibly quickly at mathematics, and it will keep getting a heck of a lot better than this. Which is hard to imagine, but likely true. And since AI is getting so powerful, I believe we need a heck of a lot more coordination for safety and alignment. Also, when Nobel laureate Sir Demis Hassabis tells me that we could cure all disease in 10 years, then I fall off the chair. Why and how? Well, at the risk of extending his argument, well, let's make disease a verifiable problem. Like mathematics. If we can do that, we might be able to cure all disease in 10 years. What a time to be alive! Alright, subscribe and hit the bell if you like this. A lot more is coming. I use Lambda to reproduce AI research papers, often in minutes. It's also great to train your own models, or fine-tune an existing one. Run inference, or text to image or video? Easy peasy. Running a DeepSeek chatbot or agent? Super fast, super reliable. 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