AI Has Solved One of Math’s $1 Million Millennium Prize Problems
The proof of a “blowup” has blown up the mathematics community.
“爆炸”的证明震惊了数学界。
On the morning of Tuesday, September 8, mathematicians at OpenAI announced that a group of 10,000 autonomous AI agents under their direction, running on an advanced model not available to the public, had found a “singularity” in the Navier-Stokes equations in three dimensions — thus resolving one of the six remaining Millennium Prize Problems posed in 2000 by the Clay Mathematics Institute, each of which carries a $1 million prize.
9 月 8 日星期二上午，OpenAI 的数学家宣布，由 10,000 个自主人工智能代理组成的小组在他们的指导下，在公众无法获得的先进模型上运行，发现了纳维-斯托克斯方程在三个维度上的“奇点”，从而解决了克莱数学研究所在 2000 年提出的剩下的六个千禧年奖问题之一，每个问题的奖金为 100 万美元。
Their result has been formally checked in the programming language Lean, giving mathematicians confidence that it is indeed correct.
他们的结果已经在精益编程语言中进行了正式检查，让数学家相信它确实是正确的。
If the result holds up to further scrutiny, it is, by a significant margin, the most important mathematical proof to have been arrived at by an artificial-intelligence model to date, possibly marking a fundamental turning point in how mathematicians tackle difficult problems.
如果结果经得起进一步审查，那么它在很大程度上是迄今为止人工智能模型得出的最重要的数学证明，可能标志着数学家解决难题的根本转折点。
This particular difficult problem deals with differential equations, which express relationships between changing quantities.
这个特别困难的问题涉及微分方程，它表达变化量之间的关系。
They are arguably the single most important mathematical tool for explaining the world around us.
它们可以说是解释我们周围世界的最重要的数学工具。
As a rule, they are easy to write down and hard to solve.
一般来说，它们很容易写下来，但很难解决。
The Navier-Stokes equations are differential equations that use Newton’s second law of motion to describe how fluids, from ocean currents to air flows, behave.
纳维-斯托克斯方程是微分方程，它使用牛顿第二运动定律来描述从洋流到空气流动的流体的行为方式。
They were first written down in the mid-19th century, and have been central to the study of fluid mechanics ever since.
它们首次写于 19 世纪中叶，从那时起就一直是流体力学研究的核心。
But one basic question about the equations has persisted: Are their solutions always well-behaved?
但关于这些方程的一个基本问题仍然存在：它们的解总是表现良好吗？
Or can their solutions evolve over time so that some infinitesimally small part of the fluid begins to flow infinitely quickly, creating a so-called singularity?
或者他们的解决方案是否会随着时间的推移而演变，从而使流体的某些无限小部分开始无限快速地流动，从而产生所谓的奇点？
The OpenAI announcement of this long-sought singularity came 12 hours after an announcement from Tristan Buckmaster at New York University that he, together with Levent Alpöge at Anthropic, had resolved several closely related problems with help from a variety of AI models, including those of OpenAI.
纽约大学的 Tristan Buckmaster 宣布，他与 Anthropic 的 Levent Alpöge 一起在包括 OpenAI 在内的各种 AI 模型的帮助下解决了几个密切相关的问题，12 小时后，OpenAI 宣布了这一期待已久的奇点。
Both AI-enabled teams relied heavily on work by Diego Córdoba of the Institute for Mathematical Sciences in Madrid and Luis Martínez-Zoroa of CUNEF University, researchers who had developed a strategy to attack the problem that radically departed from the methods most mathematicians were using.
两个人工智能团队都在很大程度上依赖于马德里数学科学研究所的迭戈·科尔多巴（Diego Córdoba）和中央财经大学的路易斯·马丁内斯·索罗亚（Luis Martínez-Zoroa）的工作，他们的研究人员制定了一种解决该问题的策略，该策略与大多数数学家使用的方法截然不同。
“I was thrilled that the problem was solved,” said Charles Fefferman of Princeton University, who wrote the Clay Institute’s official description of the Navier-Stokes problem.
“我很高兴这个问题得到了解决，”普林斯顿大学的查尔斯·费弗曼说，他撰写了克莱研究所对纳维-斯托克斯问题的官方描述。
The heroes of the story, he said, are Córdoba and Martínez-Zoroa.
他说，这个故事的英雄是科尔多瓦和马丁内斯-索罗亚。
As Buckmaster wrote in a statement announcing his results, “Let me make plain what I have said to colleagues in private: in view of this body of work, I believe Luis Martínez-Zoroa deserves a Fields Medal.”
正如巴克马斯特在宣布其研究结果的声明中所写的那样，“让我向同事们私下里说清楚的话：鉴于这项工作，我相信路易斯·马丁内斯-索罗亚应该获得菲尔兹奖。”
The Navier-Stokes equations rely on the assumption that you can zoom in on a fluid, considering endlessly smaller amounts of it.
纳维-斯托克斯方程依赖于这样的假设：您可以放大流体，同时考虑无限小的流体量。
The real world is not like this: Fluids are ultimately made of molecules and atoms.
现实世界不是这样的：流体最终是由分子和原子组成的。
They are not perfectly smooth.
它们并不完全光滑。
This means that the mathematical results about the formation of singularities don’t have any immediate practical consequences.
这意味着有关奇点形成的数学结果不会产生任何直接的实际后果。
However, those results are important because it’s surprising that such singularities are possible even in an idealized sense.
然而，这些结果很重要，因为令人惊讶的是，即使在理想化的意义上，这种奇点也是可能的。
It tells us that, straightforward as Newton’s second law appears to be, its consequences when applied to fluids are profoundly counterintuitive.
它告诉我们，尽管牛顿第二定律看起来很简单，但当它应用于流体时，其后果是非常违反直觉的。
Put another way: Turbulence is even weirder than it appears to be.
换句话说：湍流比看起来更奇怪。
The Navier-Stokes equations account for the fact that fluids can have viscosity, or friction.
纳维-斯托克斯方程解释了流体可以具有粘度或摩擦力的事实。
(Fluids with more viscosity, like honey, flow slowly, while those with less viscosity, like water, flow more quickly.) A simpler, related set of equations called the Euler equations describe fluids with zero viscosity, which flow without friction.
（粘度较大的流体，如蜂蜜，流动缓慢，而粘度较小的流体，如水，流动较快。）一组更简单、相关的方程组（称为欧拉方程）描述了粘度为零的流体，其流动时没有摩擦。
The two sets of equations are closely related — researchers often work in parallel on both.
这两组方程密切相关——研究人员经常同时研究这两组方程。
But introducing even an infinitesimal amount of friction causes a fluid to behave in a profoundly different way.
但即使引入极小的摩擦力也会导致流体的行为方式截然不同。
OpenAI’s solution is a vortex, visualized here, where yellow represents a faster rotation speed and blue slower.
OpenAI 的解决方案是一个漩涡，如图所示，其中黄色代表更快的旋转速度，蓝色代表更慢的旋转速度。
“Ten years ago, nobody believed there was a singularity for Navier-Stokes,” said Córdoba — though many believed that the Euler equations did admit a singularity.
“十年前，没有人相信纳维-斯托克斯存在奇点，”科尔多巴说，尽管许多人相信欧拉方程确实承认存在奇点。
This began to change in 2013, when Thomas Hou of the California Institute of Technology and Guo Luo , now at the Hang Seng University of Hong Kong, derived a groundbreaking result showing that the Euler equations can “blow up,” as mathematicians like to say, in a cylinder if the top and bottom halves are set spinning in opposite directions.
这种情况在 2013 年开始发生变化，当时加州理工学院的托马斯·侯 (Thomas Hou) 和现就职于香港恒生大学的郭罗 (Guo Luo) 得出了一项突破性的结果，表明如果圆柱体的上半部分和下半部分以相反的方向旋转，欧拉方程可能会“爆炸”，正如数学家常说的那样。
“That’s the first really serious claim of singularity,” Córdoba remembered.
“这是第一次真正严肃地宣称奇点，”科尔多巴回忆道。
Over the next few years, a series of results including a 2019 paper got mathematicians thinking that not only might the Euler equations have singularities, but that Navier-Stokes might as well.
在接下来的几年里，包括 2019 年一篇论文在内的一系列结果让数学家们开始思考，不仅欧拉方程可能存在奇点，纳维-斯托克斯方程也可能存在奇点。
There are a few intellectual steps to get from there to the present day.
从那时起到现在，需要采取一些智力步骤。
The first is the question of a boundary.
首先是边界问题。
The Millennium Prize version of the problem asks what happens in three-dimensional space that extends indefinitely in all directions.
该问题的千禧年奖版本询问在向各个方向无限延伸的三维空间中会发生什么。
Earlier results, like the 2013 cylinder one, posit the existence of some boundary.
早期的结果，如 2013 年第一缸的结果，假定存在某种边界。
These are useful intermediate findings, but the case without a boundary is mathematically interesting, according to Fefferman, because in models with a boundary, finding a singularity “tells you the fluid can form a singularity thanks to its interaction with the boundary — but without a boundary it’s the fluid doing the crazy stuff.”
这些都是有用的中间结果，但根据费弗曼的说法，没有边界的情况在数学上很有趣，因为在有边界的模型中，找到奇点“告诉你流体可以通过与边界的相互作用而形成奇点——但如果没有边界，流体就会做出疯狂的事情。”
The next major step involves modeling the forces that cause fluids to move.
下一个主要步骤涉及对导致流体移动的力进行建模。
These could be something as natural as gravity pulling the fluid downwards, or an artificial intervention like a propeller.
这些可能是像重力向下拉动流体一样自然的东西，也可能是像螺旋桨这样的人工干预。
Mathematicians model these forces with a technique called “forcing.” They’ve sometimes tried to introduce awkward, ungainly forcing functions to get fluids to behave in odd ways.
数学家使用一种称为“强迫”的技术对这些力进行建模。他们有时试图引入笨拙、笨拙的强制函数来让流体以奇怪的方式表现。
But the Millennium Prize version of the problem asks what can happen when the forcing function is mathematically well-behaved, or “smooth.”
但该问题的千禧年奖版本提出了当强迫函数在数学上表现良好或“平滑”时会发生什么。
In their 2013 cylinder result, Hou and Luo used computer models to simulate a scenario that might lead the Euler equations to blow up.
在 2013 年的圆柱结果中，侯和罗使用计算机模型模拟了可能导致欧拉方程崩溃的场景。
They then proved that blowup does occur by computationally accounting for all potential errors.
然后，他们通过计算考虑所有潜在错误来证明爆炸确实发生。
In the years since, similar techniques have become the dominant mode of attack on the Euler and Navier-Stokes problems.
从那时起，类似的技术已成为解决欧拉和纳维-斯托克斯问题的主要模式。
But in his 2021 doctoral dissertation, Martínez-Zoroa pioneered analytic techniques that don’t rely on computers at all.
但 Martínez-Zoroa 在 2021 年的博士论文中开创了完全不依赖计算机的分析技术。
This made him, along with Córdoba, his doctoral adviser, an oddball in the area.
这使他和他的博士生导师科尔多瓦一起成为该地区的怪人。
By 2023, the pair had proved that a version of the Euler equations with a messy forcing function displayed singularities.
到 2023 年，两人证明了具有混乱强迫函数的欧拉方程版本显示出奇点。
Córdoba likes to joke: “I don’t use AI: I have Luis.” Martínez-Zoroa added that it’s not that either of them is opposed to AI, but that “up to relatively recently, when I tried to use it for my own work, it didn’t match very well with my workflow.
科尔多巴喜欢开玩笑：“我不使用人工智能：我有路易斯。” Martínez-Zoroa 补充说，并不是他们反对人工智能，而是“直到最近，当我尝试将它用于自己的工作时，它与我的工作流程不太匹配。
I’ll have to adapt, clearly.”
显然，我必须适应。”
In broad outline, the pair’s technique relies on creating an infinite sequence of “layers,” each of which is a non-singular solution to the equation they are studying.
总的来说，两人的技术依赖于创建无限的“层”序列，每个“层”都是他们正在研究的方程的非奇异解。
(They’ve applied similar techniques to both the Euler and Navier-Stokes equations, as well as to other related systems.) They then combine those solutions in what Martínez-Zoroa calls an “infinite cascade” to produce a new solution.
（他们将类似的技术应用于欧拉方程和纳维-斯托克斯方程，以及其他相关系统。）然后，他们将这些解结合到马丁内斯-索罗亚所说的“无限级联”中，以产生一个新的解。
That new solution, they showed, contains the desired singularity.
他们表明，这个新解决方案包含了所需的奇点。
However, even though each individual layer relies on a smooth forcing function, combining them together can cause the forcing function to have undesirable mathematical properties.
然而，即使每个单独的层都依赖于平滑的强制函数，将它们组合在一起可能会导致强制函数具有不良的数学特性。
That’s why their solution fell short of satisfying the Millennium Prize criteria.
这就是为什么他们的解决方案未能满足千年奖的标准。
The remaining hurdle was to figure out how to create a similar infinite cascade that resulted not only in a singularity, but also in a smooth forcing function.
剩下的障碍是弄清楚如何创建一个类似的无限级联，不仅产生奇点，而且产生平滑的强迫函数。
That’s the step that both competing AI groups appear to have had success with.
这两个竞争的人工智能团队似乎都在这一步上取得了成功。
As Córdoba and Martínez-Zoroa were working on their techniques, AI-derived math research has been on something of a binge.
当科尔多瓦和马丁内斯-索罗亚正在研究他们的技术时，人工智能衍生的数学研究一直在进行。
Since the beginning of the summer, large language models, primarily from OpenAI and Anthropic, have been used to obtain proofs of results across many areas of math.
自今年夏天开始，大型语言模型（主要来自 OpenAI 和 Anthropic）已被用于获取许多数学领域的结果证明。
Often these results have been accompanied by formal proofs, which rely on the programming language Lean to establish, with airtight certainty, that a statement must be true.
通常，这些结果都伴随着正式的证明，这些证明依赖于编程语言精益来确定，一个陈述必须是真实的。
(The crucial bit of verification that must still be done by humans is to guarantee that the statement being shown to be true in Lean is logically equivalent to what mathematicians set out to prove.)
（验证的关键部分仍然必须由人类完成，以保证在精益中被证明是正确的陈述在逻辑上等同于数学家想要证明的内容。）
For much of the summer, that competition has seemed, if not friendly, at least not openly acrimonious.
今年夏天的大部分时间里，这种竞争即使不友好，至少也不是公开激烈的。
That changed in the last 24 hours.
过去 24 小时内情况发生了变化。
Video : This video explores why the Navier-Stokes equations are so difficult, what it means for their solutions to blow up, and how a new AI-assisted proof is reshaping one of mathematics’ most famous unsolved problems.
视频：该视频探讨了为什么纳维-斯托克斯方程如此困难，其解决方案爆炸意味着什么，以及新的人工智能辅助证明如何重塑数学最著名的未解决问题之一。
Emily Buder/ Quanta Magazine and Emily Zhang
Emily Buder/Quanta 杂志和 Emily 张
Buckmaster shared a statement just before midnight on Monday, September 7, announcing the results of his collaboration with Alpöge.
Buckmaster 在 9 月 7 日星期一午夜之前发表了一份声明，宣布了他与 Alpöge 的合作成果。
“For most of the past year progress was slow,” Buckmaster said in his statement.
“过去一年的大部分时间里，进展都很缓慢，”巴克马斯特在声明中说。
But by August 22, they had a Lean-verified proof for the Euler equations: “I can say the first LLM generated proof Levent sent me was the most horrendous I have ever read,” Buckmaster wrote.
但到了 8 月 22 日，他们得到了欧拉方程的精益验证证明：“我可以说莱文特发给我的第一个法学硕士生成的证明是我读过的最可怕的证明，”巴克马斯特写道。
The duo had planned to continue working on a more elegant write-up, but after word of their progress leaked to OpenAI, they felt they had to move up their timeline.
两人原计划继续撰写更优雅的文章，但在他们的进展消息泄露给 OpenAI 后，他们觉得必须提前制定时间表。
Buckmaster lamented that one of the three papers he and Alpöge were releasing “can only be described as AI slop.
巴克马斯特感叹道，他和 Alpöge 发布的三篇论文中的一篇“只能被描述为人工智能的废话”。
I am sorry for this.”
我对此感到抱歉。”
OpenAI admits that their work on the Navier-Stokes equation was inspired by rumors that Alpöge and Buckmaster had solved a Millennium Prize problem.
OpenAI 承认，他们对纳维-斯托克斯方程的研究受到了 Alpöge 和 Buckmaster 解决了千禧年奖问题的传言的启发。
(They hadn’t quite, though they say they have an unverified proof of blowup for a somewhat easier version of Navier-Stokes.) Using a new internal model, OpenAI employed groups of autonomous AI agents of various sizes to attack variants of the Euler and Navier-Stokes problems.
（他们并没有完全做到这一点，尽管他们说他们拥有未经验证的纳维-斯托克斯问题的爆炸证明。）使用新的内部模型，OpenAI 使用了不同规模的自主人工智能代理组来攻击欧拉和纳维-斯托克斯问题的变体。
“Nearly 100 agents worked together for approximately 50 hours to produce our Euler regularity disproof,” the company wrote in a press release.
该公司在一份新闻稿中写道：“近 100 名代理人一起工作了大约 50 个小时，以证明我们的欧拉正则性反证。”
They then deployed an even bigger group of agents — 10,000 or so — to attack Navier-Stokes.
然后，他们部署了一支规模更大的特工队伍——大约 10,000 人——来攻击纳维-斯托克斯。
After 88 hours, the agents running on the internal model had a proof of a singularity in Navier-Stokes, and after an additional 17 hours, another AI model had formalized the result.
88 小时后，在内部模型上运行的智能体获得了纳维-斯托克斯奇点的证明，又过了 17 小时，另一个人工智能模型正式确定了结果。
In total, the agents sent almost 5 million messages to each other.
代理之间总共发送了近 500 万条消息。
Sébastien Bubeck of OpenAI estimates the computational cost at several million dollars.
OpenAI 的 Sébastien Bubeck 估计计算成本为数百万美元。
As this story went to press, the details of the interaction between Buckmaster, Alpöge, and OpenAI remain murky — different parties to the conversation are presenting different versions.
截至本文发稿时，Buckmaster、Alpöge 和 OpenAI 之间的交互细节仍然不明朗——对话的不同各方提出了不同的版本。
OpenAI cedes priority for the 3D Euler result to Buckmaster and Alpöge, while claiming it for the Navier-Stokes result.
OpenAI 将 3D Euler 结果的优先权让给了 Buckmaster 和 Alpöge，同时主张 Navier-Stokes 结果的优先权。
In Buckmaster’s statement, he appears to suggest that the OpenAI researchers or their AI agents may have gained access to (and benefited from) the work that he and Alpöge did using OpenAI’s models.
在 Buckmaster 的声明中，他似乎暗示 OpenAI 研究人员或其人工智能代理可能已经获得了他和 Alpöge 使用 OpenAI 模型所做的工作（并从中受益）。
It will take some time to sort out the timelines of who did what when and to understand the mathematical significance of the new AI proofs, even if they have already been formally verified.
即使新的人工智能证明已经得到正式验证，也需要一些时间来理清谁在何时做什么的时间表，并理解新的人工智能证明的数学意义。
But regardless, the intellectual debt to Córdoba and Martínez-Zoroa seems clear.
但无论如何，科尔多瓦和马丁内斯-索罗亚在智力上的欠债似乎是显而易见的。
“I’m very happy for Tristan,” Martínez-Zoroa said.
“我为特里斯坦感到非常高兴，”马丁内斯-索罗亚说。
“It would have been nice to do this ourselves, but I’m very happy for him.”
“如果我们自己能做到这一点就好了，但我为他感到高兴。”
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《Quanta》杂志对评论进行审核，以促进知情、实质性、文明的对话。
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辱骂、亵渎、自我推销、误导、语无伦次或偏离主题的评论将被拒绝。
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