OpenAI says it has solved one of math’s "Millennium Problems"

OpenAI on Tuesday said its artificial intelligence technology has cracked one of the Millennium Problems –– seven famously difficult and unsolved questions in mathematics that have stumped even the world’s brightest minds.

In a release of its findings, the company said an internal model, not released to the public, solved the Navier-Stokes problem, a 200-year-old puzzle related to the motion of fluids.

The company said its model took 88 hours to solve the problem, and that it used as many as 10,000 “AI agents” working somewhat concurrently.

“This milestone represents substantial work by mathematicians and AI researchers. However, this is not a culmination, but rather a snapshot in time, of progress on AI development,” the company wrote in its statement.

Yet the notice has also raised a stir in the mathematics community for another reason.

The night before OpenAI’s announcement, New York University math professor Tristan Buckmaster released a statement saying he and Anthropic researcher Levent Alpöge had been looking into the Navier-Stokes problem –– a culmination of work that began last year – and had used a mix of large language models, including OpenAI’s Codex.

As they prepared to go public with their results, they say they reached out to OpenAI researchers –– and later learned researchers were about to make a similar discovery.

“I would like to be clear about what I am not claiming,” Buckmaster said in the statement. “I have not seen OpenAI’s proof. I do not know what their model did, or how. I do not know whether our data was used. I am not accusing anyone of anything.”

Addressing concerns OpenAI’s model could have used their work in progress to come to its own conclusion, OpenAI acknowledged that its work solving the Navier-Stokes problem began on September 1 after hearing what it called a rumor of another Millennium Prize problem being resolved.

It said its agents had access to tools such as the ability to read from a cached version of the internet and the ability to run code, but that it “did not see any of their work through any means until they released it publicly” and “no specific user data was accessed in order to solve this problem.”

OpenAI said it reached out to Buckmaster and Alpöge to offer them a concurrent release of results and “visibility into all of the prompts we used and to later see the poof,” according to the Tuesday statement. Alongside its release Tuesday, OpenAI also published its Lean proof, which shows how its agents cracked the problem.

The company acknowledged, however, that: “we cannot rule out that de-identified data derived from their usage of our products helped improve our models⁠ .”

Matthias Balk/picture alliance/Getty Images

White House asks OpenAI to limit its next model release

Some of the world’s most highly respected mathematicians have weighed in on OpenAI’s results –– expressing concern over how using AI to solve such complex mathematical problems can risk flattening out the field.

“It’s a little like going to watch a movie and jumping straight from the first ten minutes to the last ten minutes; technically, all the plot lines are resolved, but most of the value of the experience was lost,” Terence Tao, a math professor at the University of California, Los Angeles, told CNN.

Tao acknowledged that while it’s possible OpenAI drew on ongoing efforts from human mathematicians to solve the problem, “the dynamic is now that of frenetic competition,” he said.

“The indiscriminate use of AI is turning the subject into a meaningless production quota ‘game’ that ultimately is of very little benefit, either to mathematics or to the world,” he said.

The Navier-Stokes problem, dating back to the 19th-century work of Claude-Louis Navier and George Gabriel Stokes, involves fluid dynamics and whether these equations break down under certain conditions.

The seven “Millenium Problems” were chosen by members of The Clay Mathematics Institute of Cambridge, Massachusetts, as a collection of the most important mathematical questions of the new millennium.

In 2000, the Millennium Prize Problems was established to inspire work towards solving these complex problems, offering a $1 million prize for the first correct solution to each problem.

Only one Millenium Prize problem has been officially solved so far: the Poincaré Conjecture, a theorem about the shape of three-dimensional spaces.

In the Tuesday announcement, OpenAI said the company does not intend to claim the Millennium Prize for the result.

OpenAI 周二表示，其人工智能技术已经破解了千禧年难题之一——这七个著名的数学难题至今仍未解决，甚至难倒了世界上最聪明的头脑。

该公司在发布研究结果时表示，一个未向公众发布的内部模型解决了纳维-斯托克斯方程问题，这是一个与流体运动相关的、已有 200 年历史的难题。

该公司表示，其模型花了 88 小时才解决这个问题，并且使用了多达 10,000 个“人工智能代理”同时工作。

“这一里程碑代表了数学家和人工智能研究人员的大量工作。然而，这并非最终的成果，而只是人工智能发展进程中某一时刻的缩影，”该公司在声明中写道。

然而，该通知在数学界引起轰动还有另一个原因。

在 OpenAI 宣布这一消息的前一天晚上，纽约大学数学教授 Tristan Buckmaster 发表声明称，他和 Anthropic 研究员 Levent Alpöge 一直在研究纳维-斯托克斯问题——这是他们从去年开始的研究的最终成果——并且使用了包括 OpenAI 的 Codex 在内的多种大型语言模型。

他们表示，在准备公开研究结果时，他们联系了 OpenAI 的研究人员，后来得知研究人员即将做出类似的发现。

“我想澄清一下我没有提出什么指控，”巴克马斯特在声明中说。“我没有看到OpenAI的证据。我不知道他们的模型做了什么，也不知道他们是如何做的。我不知道他们是否使用了我们的数据。我没有指控任何人任何事。”

针对有人担心 OpenAI 的模型可能利用他们正在进行的工作得出自己的结论，OpenAI 承认，其解决纳维-斯托克斯问题的工作始于 9 月 1 日，当时他们听到了一个传言，称另一个千禧年大奖问题已被解决。

该公司表示，其特工拥有读取互联网缓存版本和运行代码等工具，但“在他们公开发布之前，该公司没有通过任何方式看到他们的任何工作”，并且“为了解决这个问题，没有访问任何特定的用户数据”。

OpenAI在周二的声明中表示，他们已联系Buckmaster和Alpöge，提出与他们同步发布结果，并“让他们了解我们使用的所有提示，以及之后的验证过程”。除了周二发布的结果外，OpenAI还公布了其Lean证明，展示了其智能体如何破解该问题。

但该公司承认：“我们不能排除从他们使用我们产品中获得的去标识化数据有助于改进我们的模型。”

Matthias Balk/picture alliance/Getty Images

白宫要求OpenAI限制其下一代模型的发布。

世界上一些最受尊敬的数学家对 OpenAI 的研究成果发表了看法——他们担心使用人工智能来解决如此复杂的数学问题可能会使该领域趋于平庸。

“这有点像去看电影，直接从前十分钟跳到最后十分钟；从技术上讲，所有的故事情节都得到了解决，但大部分的观影体验价值都丧失了，”加州大学洛杉矶分校的数学教授陶哲轩告诉 CNN。

陶哲轩承认，虽然 OpenAI 有可能借鉴了人类数学家们为解决这个问题所做的持续努力，但他表示，“现在的动态是一场激烈的竞争”。

他说：“人工智能的滥用正在将数学变成一场毫无意义的生产配额‘游戏’，最终对数学或世界都几乎没有任何益处。”

纳维-斯托克斯问题可以追溯到 19 世纪克劳德-路易·纳维和乔治·加布里埃尔·斯托克斯的工作，它涉及流体动力学以及这些方程在某些条件下是否会失效。

这七个“千禧年难题”是由马萨诸塞州剑桥市克雷数学研究所的成员选出的，它们代表了新千年最重要的数学问题。

2000 年，千禧年难题设立，旨在激励人们努力解决这些复杂的难题，并为每个难题的第一个正确答案提供 100 万美元的奖金。

迄今为止，只有一项千禧年大奖难题得到了正式解决：庞加莱猜想，这是一个关于三维空间形状的定理。

OpenAI在周二的声明中表示，该公司不打算就此成果申领千禧奖。