我们正在分享纳维-斯托克斯存在性与光滑性问题的一个解决方案，这是千禧年大奖难题之一。这一证明由OpenAI内部系统生成，表明流体运动的纳维-斯托克斯方程动力学可以在有限时间内发展出奇点。我们同时分享证明的书面说明和Lean中的形式化验证。
千禧年大奖难题(在新窗口中打开)代表了数学前沿一些最深奥的问题。三维光滑流体运动是否会破裂的问题，至今已悬而未决约90年。
我们工作的一个主要目标是赋能科学家，推动有益于全人类的研究和技术进步。为了解决纳维-斯托克斯问题，我们使用了一个内部模型，其能力显著超过GPT‑6 Astra。我们认为，向世界通报AI进展的步伐以及对即将推出的模型的预期，是非常重要的。
问题
纳维-斯托克斯方程利用牛顿第二运动定律（“F=ma”）来描述流体的运动方式。重要的是，它将流体视为连续介质，而不是追踪单个分子。这些方程用于飞机设计、天气预报和血流研究。
这些动力学方程的一个基本开放问题是，流体的连续介质近似是否会破裂。具体而言，对于具有恒定密度的三维不可压缩流体，即使运动从光滑状态开始，纳维-斯托克斯方程是否可能发展出“奇点”？这里，奇点意味着动力学导致流体中的速度在有限时间内无界增长。奇点的发展必须在存在黏性（倾向于平滑运动）的情况下发生。由于真实流体不可能以无限速度运动，这将标志着方程对流体建模方式的失效。要继续对系统建模，就需要逐一追踪每个粒子的行为。
这些方程可追溯至十九世纪克劳德-路易·纳维和乔治·加布里埃尔·斯托克斯的工作。1934年，让·勒雷证明了广义意义上的解存在，但解是否始终保持光滑成为核心未解问题。2000年，克莱数学研究所将纳维-斯托克斯存在性与光滑性问题列为七个千禧年大奖难题之一。
结果
我们的系统生成了一个解析证明和Lean形式化验证，表明一个初始光滑的静止流体可以在有限时间内发展出奇点。该流体受到光滑外力的作用，其能量在整个动力学过程中（从静止到奇点形成）保持有限。这通过确立官方千禧年问题表述(在新窗口中打开)中的陈述“C”（以及“D”），解决了纳维-斯托克斯千禧年大奖难题。
该解是一个涡旋，即旋转的流体漩涡，向内螺旋并逐渐拉长，形似意大利面。这个中心区域在收缩的同时加速，其能量仍保持有限，符合物理定律的要求。技术挑战在于，方程需要通过流体自身的运动发展出破裂，而不是例如我们手动施加无限大的力。更数学化地说，纳维-斯托克斯方程中描述运动的项——加速度、压力梯度、动量传递、黏性——必须既变得_大_，又以精确方式_相互抵消_。这种精细平衡留下了一个光滑的外力，即使流体速度无界增长。
局部不可压缩运动的快照。橙色表示较快的角旋转；青色表示较慢的旋转。循环速度也取决于半径。轨迹显示向内螺旋和轴向拉伸。
我们如何找到证明
自8月28日起，我们一直在训练一个新的内部模型，该模型在我们的基准测试中表现出前所未有的性能，包括数学领域。该模型的训练仍在进行中，其性能持续提升。
9月1日（星期二），我们听到传言称两个千禧年大奖难题已被解决。受这些传言以及我们内部模型性能阶跃式提升的启发，我们启动了一项工作，在所有开放的千禧年大奖难题和其他几个高影响力问题上对其进行评估。
我们使用了一个由内部模型驱动的协调代理系统。代理可以访问工具，例如从互联网缓存版本中读取信息以及运行代码。代理被细分为多个小组，组内可以通信。小组规模各异，产生纳维-斯托克斯解决方案的小组涉及约10,000个并发代理。我们始终维持与所有前沿模型评估相同的严格保障措施，包括监控和隔离。
对于每个问题，我们向不同的代理组提示问题陈述的不同变体，覆盖问题的所有变体。对于纳维-斯托克斯问题，我们向不同的代理组建议了版本“A”和“B”（纳维-斯托克斯问题的特定形式，可能导致证明）以及版本“C”和“D”（可能导致反证）。
除了完整的千禧年大奖难题外，我们还要求多代理系统尝试一组“较简单”的问题。其中一个问题是纳维-斯托克斯问题在移除黏性项后的极限情况下的类似爆破问题。这被称为欧拉方程的正则性问题，我们的代理通过解决该问题让我们感到惊讶。他们解决的具体变体是_无外力_版本，即没有外力施加于流体。近100个代理协同工作约50小时，生成了我们的欧拉正则性反证。1
一旦我们看到欧拉解，我们认为纳维-斯托克斯是最有希望攻克的问题。因此，我们决定将资源集中于纳维-斯托克斯。为此，我们将代理从其他千禧年问题中转移出来，并向这些代理提示欧拉解。当我们的内部模型在努力过程中推出进一步训练的版本时，我们将代理更新为该模型。
我们鼓励不同的代理组探索多样化的方法。一段时间后，我们使用Codex整合每个代理组最有用的见解，对代理组进行交叉授粉。这些后续提示利用了代理自身的中间结果。找到纳维-斯托克斯解决方案的小组就是以这种方式得到引导的。
这些智能体于9月5日（星期六）达成解决方案，距首批智能体启动约88小时。通过GPT‑6 Astra进行的Lean形式化与验证又额外耗时17小时。
在所有尝试解决的问题中，智能体共发送了490万条消息，使用了约3000亿个输出令牌。在解决纳维-斯托克斯问题的过程中，智能体发送了270万条消息，使用了约1300亿个输出令牌。
并行工作
我们的工作始于9月1日，当时我们听到一则传言，后来意识到这与Anthropic员工Levent Alpöge及纽约大学数学教授Tristan Buckmaster有关。在我们完成整个项目及Lean验证（9月6日）后，基于传言认为他们也可能已获得纳维-斯托克斯问题的解，我们主动联系他们，提议并行发布我们的结果，并在联合公告中承认他们的优先权。当时我们得知他们解决的是受迫欧拉问题。在这些讨论中，我们向他们提供了我们使用的所有提示词，并随后展示了证明。我们承认他们在受迫欧拉问题上的优先权，并祝贺他们取得的卓越数学成就。
我们（研究人员及智能体）在他们公开发布之前，未通过任何途径看到他们的工作——特别是，为解决此问题未访问任何特定用户数据。尽管可能性不大，但我们无法排除源自他们使用我们产品所得到的去标识化数据可能帮助改进了我们的模型。然而，我们的证明存在显著差异，甚至在欧拉问题情形下，所证明的精确结果也各不相同（受迫与不受迫之别）。
进展与责任
我们发布此结果的目标是报告我们AI模型的实质性进展。我们无意就此结果申领千禧年大奖。
这一里程碑代表了数学家与AI研究者的大量工作。然而，这并非终点，而是AI发展进程中的一个时间快照。
我们相信，我们现已进入AI进步的下一个阶段，今日的结果为此提供了进一步证据。我们正专注于理解这一模型，并利用所学来指导并把握我们追求能力进一步突破的节奏。我们的关键目标之一是构建可引导、可问责且与人类紧密相连的AI系统，这可能需要在进展节奏上做出更为审慎的选择，同时我们继续履行确保AGI惠及全人类的使命。
We’re sharing a solution to the Navier–Stokes existence and smoothness problem, one of the Millennium Prize Problems. This proof, produced by an internal OpenAI system, shows that the dynamics of the Navier-Stokes equations for fluid motion can develop a singularity in finite time. We’re sharing both a writeup of the proof and a formalization in Lean.
The Millennium Prize Problems(opens in a new window) represent some of the deepest questions at the frontier of mathematics. The question of whether smooth three-dimensional fluid motion can break down has remained unresolved for roughly 90 years.
A major goal of our work is to empower scientists to advance research and technology that benefits all of humanity. To solve the Navier–Stokes problem, we used an internal model that is significantly more capable than GPT‑6 Astra. We believe it is important to inform the world about the pace of AI progress and what to expect from upcoming models.
The problem
The Navier–Stokes equations use Newton’s second law of motion (“F=ma”) to describe how fluids move. Importantly, they treat a fluid as a continuous medium rather than tracking individual molecules. These equations are used for aircraft design, weather forecasting, and the study of blood flow.
A fundamental open question for these dynamical equations has been whether the continuum approximation of the fluid can break down. Specifically, can the Navier–Stokes equations for a three-dimensional incompressible fluid with constant density develop a “singularity,” even when the motion starts smoothly? Here, a singularity means the dynamics lead to speeds in the fluid growing without bound within a finite amount of time. The development of a singularity would have to happen despite the presence of viscosity, which tends to smooth out motion. Because a real fluid cannot move infinitely fast, this would mark a breakdown in how the equations model the fluid. To continue modeling the system, one would then need to track the behaviour of each particle individually.
The equations date to the nineteenth-century work of Claude-Louis Navier and George Gabriel Stokes. In 1934, Jean Leray proved that solutions exist in a generalized sense, but whether they always remain smooth became a central unanswered question. In 2000, the Clay Mathematics Institute named the Navier–Stokes existence and smoothness problem one of seven Millennium Prize Problems.
The result
Our system produced an analytical proof and a Lean formalization that an initially smooth fluid at rest can develop a singularity in a finite time. The fluid has a smooth force applied to it, and its energy remains finite through the entire dynamics, from rest to the formation of the singularity. This resolves the Navier–Stokes Millennium Prize problem by establishing statement “C” (and also “D”) in the official Millennium Prize formulation(opens in a new window).
The solution is a vortex, a spinning swirl of fluid, that spirals inward and gets increasingly elongated, like spaghetti. This central region shrinks while it speeds up in such a way that its energy still stays finite, as required by the laws of physics. The technical challenge is for the equations to develop the breakdown through the motion of the fluid itself, rather than, for example, us putting in an infinite force by hand. More mathematically, the terms in the Navier–Stokes equations that describe the motion—acceleration, pressure gradients, momentum transfer, viscosity—must both become big yet cancel in a precise way. This detailed balance leaves a smooth external force even as the velocity of the fluid grows without bound.
A snapshot of local incompressible motion. Orange marks faster angular rotation; teal marks slower rotation. Circulating speed also depends on radius. The trajectories show inward spiraling and axial stretching.
How we found the proof
Since August 28 we have been training a new internal model that has exhibited unprecedented performance in our benchmarks, including mathematics. This model’s training is ongoing and its performance continues to improve.
On Tuesday, September 1, we heard rumors that two Millennium Prize problems had been resolved. Inspired by these rumors and by the step change in performance of our internal model, we launched an effort to evaluate it on all open Millennium Prize problems and a few other high-impact problems.
We used a system of coordinating agents powered by our internal model. The agents had access to tools such as the ability to read from a cached version of the internet and the ability to run code. Agents were subdivided into groups with the ability to communicate within the group. The groups varied in size, and the group that produced the Navier–Stokes resolution involved on the order of 10,000 concurrent agents. At all times we maintained the same strict safeguards that we apply to all our frontier model evaluations, including monitoring and isolation.
For each problem, we prompted different groups of agents with different variants of the problem statement, covering all variants of the problem. For the Navier–Stokes problem, we suggested versions “A” and “B” (particular forms of the Navier–Stokes problem which would result in a proof) and versions “C” and “D” (which would result in a disproof) to separate groups of agents.
In addition to the full Millennium Prize problems, we asked our multiagent system to try a set of “easier” problems. One of these problems was a similar blowup question for the limit of the Navier–Stokes problem with the viscosity term removed. This is known as the regularity problem for the Euler equations, and our agents surprised us by resolving this question. The specific variant of the question that they resolved was the unforced version, where no external force is applied to the fluid. Nearly 100 agents worked together for approximately 50 hours to produce our Euler regularity disproof.1
Once we saw the Euler solution, we thought that Navier–Stokes was the most promising problem to work on. Thus, we decided to devote our resources to Navier–Stokes. To do so, we shifted agents away from the other Millennium Problems and prompted these agents with the Euler resolution. When a further trained version of our internal model became available over the course of the effort, we updated our agents to that model.
We encouraged different groups of agents to explore a diversity of approaches. After some time, we cross-pollinated the agent groups by using Codex to consolidate the most useful insights from each agent group. These follow-up prompts drew on the agents’ own intermediate results. The group that found the solution to Navier–Stokes was guided in such a way.
The agents arrived at their resolution on Saturday, September 5, about 88 hours after the first agents were launched. Lean formalization and verification took an additional 17 hours via GPT‑6 Astra.
Across all attempted problems, the agents sent 4.9 million messages and used about 300 billion output tokens. In the process of resolving the Navier–Stokes problem, the agents sent 2.7 million messages and used approximately 130 billion output tokens.
Concurrent work
Our effort began on September 1st after hearing a rumor which we later realized was related to Levent Alpöge, an Anthropic employee, and Tristan Buckmaster, a math professor at NYU. After the completion of our full project and Lean verification (on September 6th), believing from the rumor they also had a solution of Navier–Stokes, we reached out to them to offer a concurrent release of our result and to recognize their priority in a joint announcement. At that point we found out that they had a resolution of the forced Euler problem. In these discussions we offered them visibility into all of the prompts we used and later to see the proof. We recognize the priority of their work on forced Euler and congratulate them on their remarkable mathematical achievement.
We (the researchers and the agents) did not see any of their work through any means until they released it publicly — in particular, no specific user data was accessed in order to solve this problem. While unlikely, we cannot rule out that de-identified data derived from their usage of our products helped improve our models. However, our proofs differ significantly and even the precise results proved are different in the Euler case (forced vs unforced).
Progress and responsibility
Our goal in releasing this result is to report on the substantial progress of our AI models. We do not intend to claim the Millennium Prize for this result.
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.
We believe we are now in the next period of AI progress, and today’s results provide further evidence of this. We are focusing on understanding this model, and using what we learn to help us guide and pace how we pursue further advances in capability. One of our key goals is to build AI systems which are steerable, accountable, and connected to people, which may require more deliberate choices about the pace of progress, as we continue our mission to ensure AGI benefits all of humanity.
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