From CFD to AI: My Take on the Navier–Stokes Breakthrough

Created on September 09, 2026

2026

I woke up groggily at five this morning, intending to check my messages and go straight back to sleep. Then I saw the news, and any hope of getting back to sleep vanished. I spent forty minutes going through OpenAI’s presentation, then asked GPT to summarize the original paper for me. By the time I finished reading, it felt as though my whole world had exploded. My doctoral research was in CFD; I dealt with the Navier–Stokes equations every day. Later, I “defected” to the AI camp. So this breakthrough hit particularly close to home.

The key numbers: 10,000 agents, 130 billion tokens, 88 hours to find a counterexample, and another 17 hours to verify it. They first found a solution without the viscosity term. Only after realizing that a complete resolution of the general case might be within reach did they decide to go all in. The counterexample turned out to be a stretched vortex structure, which makes physical sense to me. If I had been asked to guess, that is the kind of structure I would have guessed too.

What makes constructing a counterexample so difficult? The Navier–Stokes equations contain several terms—primarily the time derivative, momentum transport, pressure gradient, and viscosity. When a singularity develops in the velocity field, these terms blow up. The challenge is to design an initial field so precisely that the sum of these diverging terms remains finite over a period of time. It requires extraordinarily delicate construction. To me, it feels a little like stacking eggs: one tiny misstep, and all your work comes crashing down.

Has the problem been completely solved? Not yet. The most general case, as I understand it, should have no external forcing; this result still involves a smooth external force. I do not specialize in the mathematical structure of the Navier–Stokes equations, so I hesitate to make sweeping claims. Still, my intuition is that removing the forcing should not be all that difficult. Otherwise, why would solving the smoothly forced case also qualify for the million-dollar prize?

Will this have a huge impact on CFD? No—at least not on engineering practice. I see two main reasons:

  1. We do not encounter such extreme flow fields in everyday applications.
  2. Using the Navier–Stokes equations presupposes the continuum assumption. For a flow with this kind of singularity, that assumption would already have broken down, so the equations would no longer be applicable in the first place.

The impact, then, is mainly on mathematics. Those of us in CFD can express our astonishment—and then get back to coding, debugging, analyzing flow fields, and writing papers.

A friend asked me when humanity would finally throw off the coffin lid and rise again. My answer was: “Tired, aren’t we? Might as well lie back down.” I switched sides and joined the AI camp because I could see where things were heading. Yet I still feel powerless. If I cannot even manage to be a faithful servant of AI, then let it all come crashing down—the sooner, the better. On this downhill stretch, what we need is not a foot on the brake, but a foot on the accelerator.

中文原文 · Chinese Original

CFD转行AI选手谈谈这次的NS难题突破

今天早上5点迷迷糊糊醒来，本来想看一眼消息再睡，没想到看到这则新闻，直接睡意全无，花了40分钟看了openai的界面，然后让gpt帮我总结了一下原论文，看完感觉整个世界都爆炸了。因为我博士就是搞CFD的，天天和NS方程打交道，后来“投敌”加入了AI阵营，所以这次的突破对我的触动尤其大。

关键信息：10000个agents，1300亿tokens，88小时找反例 + 17小时验证；一开始先找到的不带粘性项的解，意识到有可能能彻底解决一般形式，才决定all-in的。找到的反例是一个拉伸的涡结构，倒是符合物理直觉，让我猜的话我也猜这个结构。

构造反例的难点在哪里？NS方程有好几项（主要是时间导数，动量输运，压力梯度，粘性这四项），在速度场出现奇异点的时候，各项都会变成无穷大，现在要精细地设计这么一个初始场，使得几项无穷大之和在一段时间内一直保持有限，需要非常精细的设计，感觉有点像叠鸡蛋，稍微不小心就前功尽弃。

彻底解决这个问题了吗？还没有。最一般的形式应该是无外力的，这次求解的还是有光滑外力的。我不是专业研究NS方程结构的，不敢随意下定论，不过我感觉，把外力去掉应该也不是什么太难的问题，不然为啥能解决光滑外力的情况也能拿百万奖金呢？

会对CFD界有巨大影响吗？不会，至少对于工程界不会，主要原因有二：1、现实生活中碰不到这么极端的流场；2、NS方程的使用有先决条件——连续性假设，对于这种有奇点的流场，早就不适用这个假设了，也自然不可能拿NS方程求解。所以，这次的影响主要还是针对数学界，CFD界只能表示震惊，然后继续写代码/debug/分析流场/写论文。

有朋友问我，啥时候人类能揭棺而起，我的回答是“累了吧，直接躺平吧”。我也是看到AI发展的大势才跳反加入的AI阵营，然而现在依然感到无力，如果连AI忠实的仆人都做不好，那就索性毁灭得越快越好吧。下坡期，我们需要的不是一脚刹车，而是一脚油门。