Using AI, Mathematicians Find Hidden Glitches in Fluid Equations

Samuel Velasco/ Quanta Magazine

塞缪尔·贝拉斯科/广达杂志

Nearly 200 years ago, the physicists Claude-Louis Navier and George Gabriel Stokes put the finishing touches on a set of equations that describe how fluids swirl.

大约 200 年前，物理学家克劳德·路易斯·纳维尔和乔治·加布里埃尔·斯托克斯完成了一组描述流体如何旋转的方程。

And for nearly 200 years, the Navier-Stokes equations have served as an unimpeachable theory of how fluids in the real world behave — from ocean currents threading their way between the continents to air wrapping around an aircraft’s wings.

近 200 年来，纳维-斯托克斯方程一直是现实世界中流体行为的无可挑剔的理论——从大陆之间穿行的洋流到飞机机翼周围的空气。

Nevertheless, many mathematicians suspect that glitches hide deep within the equations.

然而，许多数学家怀疑方程深处隐藏着故障。

They have a hunch that in certain situations, the theory fails.

他们有预感，在某些情况下，该理论会失败。

In these cases, the equations will predict a fluid moving in some unphysical, incomprehensible way — spinning into an impossibly fast vortex, for instance, or instantly reversing its flow.

在这些情况下，方程将预测流体以某种非物理的、难以理解的方式移动——例如旋转成一个难以置信的快速漩涡，或者立即逆转其流动。

Some quantity in the equations will grow infinitely large, or “blow up,” as mathematicians put it.

方程中的某些量会变得无限大，或者用数学家的话说“爆炸”。

One of the greatest challenges in math is to understand the equations that describe a river current or a breeze.

数学中最大的挑战之一是理解描述河水或微风的方程。

Researchers are tackling the problem with a new computational approach.

研究人员正在用一种新的计算方法来解决这个问题。

Despite immense effort, no one has been able to come up with a situation where the Navier-Stokes equations falter.

尽管付出了巨大的努力，但没有人能够提出纳维-斯托克斯方程失效的情况。

Doing so — or, alternatively, proving that the equations never blow up — would come with a $1 million reward.

这样做——或者，证明方程永远不会崩溃——将获得 100 万美元的奖励。

And so, as a prelude to solving the Navier-Stokes problem, mathematicians have searched for blowups (also called singularities) in an assortment of simplified fluid equations, such as those that operate in only one dimension.

因此，作为解决纳维-斯托克斯问题的前奏，数学家们在各种简化的流体方程中寻找爆炸（也称为奇点），例如仅在一维中运行的方程。

They’ve found them.

他们找到了。

But essentially all the singularities they’ve identified have been “stable,” meaning that they can form in many possible ways.

但基本上他们发现的所有奇点都是“稳定的”，这意味着它们可以以多种可能的方式形成。

In the most realistic fluid theories, including Navier-Stokes, blowups (if they exist) are likely to be far more delicate, occurring in an unimaginably precise way.

在包括纳维-斯托克斯在内的最现实的流体理论中，爆炸（如果存在的话）可能要微妙得多，以难以想象的精确方式发生。

These “unstable” blowups have been nearly impossible to find, the ultimate needles in the haystack.

这些“不稳定”的爆炸几乎不可能被发现，就像大海捞针一样。

In these realistic theories, “a lot of people believe that there are singularities, but that they are unstable, so we never see them,” said Charlie Fefferman , a mathematician at Princeton University who formulated the million-dollar Navier-Stokes challenge.

在这些现实的理论中，“很多人相信存在奇点，但它们不稳定，所以我们永远看不到它们，”普林斯顿大学数学家查理·费弗曼（Charlie Fefferman）说，他提出了价值百万美元的纳维-斯托克斯挑战。

Now one group of mathematicians has developed a way of training machines to spot these phantom glitches.

现在，一组数学家已经开发出一种训练机器来发现这些幽灵故障的方法。

In a preprint posted in September, they reexamined simpler fluid equations already known to host a stable singularity.

在 9 月份发布的预印本中，他们重新检查了已知存在稳定奇点的更简单的流体方程。

There, they found additional potential blowup scenarios — including unstable ones.

在那里，他们发现了其他潜在的爆炸场景——包括不稳定的场景。

It was the first time a possible unstable singularity was uncovered in a fluid of more than one dimension.

这是第一次在多维流体中发现可能不稳定的奇点。

The team went on to discover an assortment of unstable singularity candidates in several other fluid equations as well.

该团队还在其他几个流体方程中发现了各种不稳定奇点候选者。

They have not found any million-dollar singularities.

他们还没有发现任何价值百万美元的奇点。

And they still need to rigorously prove that the ones they have found do indeed blow up.

他们还需要严格证明他们发现的那些确实会爆炸。

But their success in uncovering potential unstable singularities in simple models raises hopes that it will also be possible to find unstable blowups in higher-stakes scenarios.

但他们成功地发现了简单模型中潜在的不稳定奇点，这让人们燃起了希望，即在更高风险的场景中也有可能发现不稳定的爆炸。

“The idea of an unstable singularity no longer prevents the discovery of the singularity,” said Fefferman, who was not involved in the new research.

“不稳定奇点的想法不再妨碍奇点的发现，”未参与这项新研究的费弗曼说。

A solution to the Navier-Stokes equations captures a slice of eternity.

纳维-斯托克斯方程的解捕捉到了永恒的一部分。

Solving the equations for some initial state of the fluid will tell you the fluid’s velocity at each point in space and at every moment in time.

求解流体的某些初始状态的方程将告诉您流体在空间中每个点和时间上每个时刻的速度。

In one simple solution, a fluid might start calm and remain calm forever.

在一个简单的解决方案中，流体可能会开始平静并永远保持平静。

In a more complicated setup, gentle currents might merge into whirlpools and eddies.

在更复杂的设置中，温和的水流可能会合并成漩涡和漩涡。

The great mystery is whether every solution — every single possible fluid history that satisfies the Navier-Stokes equations — makes sense everywhere and always.

最大的谜团是，每一个解——满足纳维-斯托克斯方程的每一个可能的流体历史——是否在任何地方、任何时候都有意义。

But tackling the Navier-Stokes equations for fluids in three dimensions is unspeakably difficult, so mathematicians have started with easier versions of the problem.

但解决三维流体的纳维-斯托克斯方程是难以言喻的困难，因此数学家们开始研究该问题的更简单的版本。

For instance, the Euler equations assume that fluids flow with no internal friction, or viscosity.

例如，欧拉方程假设流体流动时没有内摩擦或粘度。

Energy doesn’t dissipate in these frictionless fluids, so they should blow up more easily than viscous ones.

这些无摩擦流体中的能量不会消散，因此它们应该比粘性流体更容易爆炸。

But even in this simpler scenario, finding a blowup solution is hard.

但即使在这种更简单的情况下，找到爆炸解决方案也很困难。

Fluid equations are generally too complicated to solve directly with pencil and paper.

流体方程通常太复杂，无法直接用铅笔和纸求解。

So a common approach is to use a computer to simulate the fluid’s motion and get an approximate sense of the conditions that seem to produce a blowup.

因此，一种常见的方法是使用计算机模拟流体的运动，并大致了解似乎会产生爆炸的条件。

If you can narrowly identify the blowup-producing conditions, you might be able to use that knowledge to rigorously prove that a blowup truly exists.

如果您能够精确地识别爆炸产生的条件，您也许能够利用这些知识来严格证明爆炸确实存在。

That’s the approach that Thomas Hou and Guo Luo took in 2013, when they simulated a digital liquid in a can.

这就是 Thomas Hou 和郭罗在 2013 年采用的方法，当时他们模拟了罐中的数字液体。

They set the top half of the liquid spinning in one direction and the bottom half in the other, then evolved this scenario through time using the Euler equations.

他们将液体的上半部分设置为向一个方向旋转，下半部分设置为另一个方向，然后使用欧拉方程随着时间的推移演变这种情况。

Eventually, at points where the opposing flows met along the can’s boundary, the vorticity (a measure of how much the liquid spins around a point) got big — bigger than their computer could handle.

最终，在相反的流动沿着罐子边界相遇的点处，涡度（衡量液体围绕一个点旋转多少的量度）变得很大——比他们的计算机能够处理的还要大。

This was a hint that a similar set of conditions would lead to a blowup.

这暗示着类似的条件会导致爆炸。

But it was not a guarantee.

但这并不能保证。

“The graveyards are strewn with alleged singular solutions of 3D Euler,” Fefferman said.

“墓地里到处都是所谓的 3D 欧拉奇异解，”费弗曼说。

It took Hou and another collaborator, Jiajie Chen , nearly a decade to remove the “alleged.” In 2022, they used a computer to prove that the singularity candidate implied the existence of a true singularity .

侯和另一位合作者陈家杰花了近十年的时间才删除“指控”。 2022年，他们用计算机证明了奇点候选者暗示了真正奇点的存在。

It was a landmark proof, and it got mathematicians hungry to push the frontier even further.

这是一个具有里程碑意义的证明，它让数学家们渴望进一步推动前沿。

The research depended on computer simulations, which meant that tiny adjustments to the initial state of the digital fluid (or any digital rounding errors) wouldn’t affect the fluid’s fate.

该研究依赖于计算机模拟，这意味着对数字流体初始状态（或任何数字舍入误差）的微小调整不会影响流体的命运。

A singularity would still occur at the can’s boundary even if things played out a little bit differently.

即使事情的发展略有不同，奇点仍然会出现在罐子的边界处。

Because of this, the singularity was stable.

因此，奇点是稳定的。

But a singularity need not be stable.

但奇点不一定是稳定的。

A blowup might occur only when the fluid is set up in the most delicate of ways.

只有当流体以最微妙的方式形成时，才可能发生爆炸。

In such a case, any adjustment to that initial arrangement, no matter how small, would prevent the fluid from blowing up.

在这种情况下，对初始布置的任何调整，无论多小，都会防止流体喷出。

Many mathematicians conjecture that if singularities do lurk in more realistic fluid equations, they’ll be unstable like this, springing up without warning.

许多数学家推测，如果奇点确实潜伏在更现实的流体方程中，它们就会像这样不稳定，毫无预警地突然出现。

They’ll also be far harder to find.

它们也将更难找到。

It’s essentially impossible to track down an unstable singularity candidate with a computer simulation.

通过计算机模拟来追踪不稳定奇点候选者基本上是不可能的。

First you’d need a cosmic stroke of luck to land on exactly the right initial configuration for your fluid — akin to trying to balance a pen perfectly on its tip, said Tristan Buckmaster , a mathematician at New York University.

纽约大学数学家特里斯坦·巴克马斯特 (Tristan Buckmaster) 表示，首先，你需要运气好才能为你的液体找到完全正确的初始配置，就像试图在笔尖上完美平衡一支笔一样。

Then, to keep it balanced, you’d also have to evolve the fluid flawlessly from one moment to the next, since even the smallest deviation will tip it onto a path that doesn’t blow up.

然后，为了保持平衡，你还必须从一个时刻到下一个时刻完美地演化流体，因为即使是最小的偏差也会使它倾斜到一条不会爆炸的路径上。

Computers aren’t capable of infinite precision.

计算机无法达到无限的精度。

They’ll inevitably introduce numerical errors that, though tiny, will stop the unstable singularity from forming.

它们将不可避免地引入数值误差，虽然很小，但会阻止不稳定奇点的形成。

“It’s like the wind blowing on your pen,” Buckmaster said.

“这就像风吹在你的笔上，”巴克马斯特说。

A frictionless fluid spinning against a cylinder can “blow up,” according to a landmark proof from Thomas Hou (left) and Jiajie Chen in 2022.

根据 Thomas Hou（左）和 Jiajie Chen 在 2022 年做出的具有里程碑意义的证明，在圆柱体上旋转的无摩擦流体可能会“爆炸”。

Vicki Chiu; Courtesy of Jiajie Chen

赵薇;由陈家杰提供

As a result, almost all blowup candidates have been stable.

结果，几乎所有的爆炸候选者都保持稳定。

So Buckmaster and his colleague Ching-Yao Lai , now at Stanford University, began to work out a potentially wind-proof way of finding unstable ones.

因此，巴克马斯特和他现在在斯坦福大学的同事赖清耀开始研究一种可能防风的方法来寻找不稳定的物体。

They didn’t set out to do so.

他们并没有打算这样做。

In 2021, they used a neural network as a new way to search indiscriminately for singularity candidates of any kind.

2021 年，他们使用神经网络作为一种新方法来不加区别地搜索任何类型的候选奇点。

A neural network is, in general, a function defined by a vast array of numbers.

一般来说，神经网络是由大量数字定义的函数。

These numbers get carefully adjusted through a highly efficient “training” process of guessing, checking, and refining until the function can perform some desired task.

这些数字通过猜测、检查和细化的高效“训练”过程进行仔细调整，直到函数可以执行某些所需的任务。

For instance, if you calibrate a neural network using thousands of labeled photos of cats and dogs, it “learns” to take in unlabeled images it’s never seen before and label them “cat” or “dog.”

例如，如果您使用数千张带标签的猫和狗的照片来校准神经网络，它会“学习”接收以前从未见过的未标记图像，并将它们标记为“猫”或“狗”。

Buckmaster and Lai turned to what’s known as a physics-informed neural network, or PINN.

Buckmaster 和 Lai 转向了所谓的物理信息神经网络（PINN）。

Unlike an image-classifying neural network, a PINN doesn’t learn by studying external data.

与图像分类神经网络不同，PINN 不会通过研究外部数据来学习。

Instead, it takes a partial differential equation — an equation that describes how a system changes over time — and adjusts itself until it can represent a function that solves that equation.

相反，它采用偏微分方程（描述系统如何随时间变化的方程）并进行自我调整，直到它可以表示求解该方程的函数。

It can, for instance, take fluid equations and train itself to home in on a function that captures a valid history of a fluid, possibly one that contains a singularity.

例如，它可以采用流体方程并训练自己寻找一个捕获流体有效历史的函数，该流体可能包含奇点。

There’s no time, so you don’t care that it’s unstable.

没有时间，所以你不在乎它不稳定。

Tristan Buckmaster, New York University

特里斯坦·巴克马斯特，纽约大学

But no computer technique can directly render the infinite nature of a singularity.

但没有任何计算机技术可以直接呈现奇点的无限本质。

Imagine playing the simulation of your fluid and watching it move forward in time.

想象一下，玩流体模拟并观察它随着时间的推移向前移动。

You might represent some quantity, such as the velocity at different points in the fluid, as a curve on a graph.

您可以将一些量（例如流体中不同点的速度）表示为图形上的曲线。

As the fluid changes over time, you’ll see that curve change as well, like a movie.

随着流体随着时间的推移而变化，您也会看到曲线发生变化，就像电影一样。

If the curve gets much steeper from one frame to the next, the fluid might be approaching a singularity.

如果曲线从一帧到下一帧变得更加陡峭，则流体可能正在接近奇点。

The simulation, however, can’t reach that final destination.

然而，模拟无法到达最终目的地。

The computer will run out of memory before the curve gets infinitely steep, crashing the program.

在曲线变得无限陡峭之前，计算机将耗尽内存，从而导致程序崩溃。

Then you can’t know for sure what was going to happen — if you were truly headed for a blowup or not.

然后你就无法确定会发生什么——是否真的会发生爆炸。

To sidestep the inconvenience of infinity, mathematicians have recently focused their search on singularities with a special property called self-similarity.

为了避免无穷大带来的不便，数学家最近将搜索重点放在具有称为自相似性的特殊属性的奇点上。

This means that there is a way of stretching the velocity curve in one frame to match the steeper velocity curve in a later frame.

这意味着有一种方法可以拉伸一帧中的速度曲线以匹配后面帧中更陡峭的速度曲线。

And so, if you want to catch a potential singularity, you no longer need to try to watch the curve get infinitely steep.

因此，如果你想捕捉潜在的奇点，你不再需要尝试观察曲线变得无限陡峭。

Instead, you can zoom in on the steepening section of the curve while the movie plays in a way that neutralizes the steepening.

相反，您可以放大曲线的陡峭部分，同时以中和陡峭的方式播放影片。

From this new, dynamic perspective, the curve gets closer and closer to a frozen curve of finite steepness instead.

从这个新的动态角度来看，曲线越来越接近有限陡度的冻结曲线。

This transformation renders the target — the frozen limit — a finite object that a finite computer can handle.

这种转换将目标（冻结极限）呈现为有限计算机可以处理的有限对象。

Buckmaster and Lai realized that PINNs could be an extremely efficient way of finding these frozen solutions to fluid equations.

Buckmaster 和 Lai 意识到 PINN 可能是寻找流体方程的冻结解的极其有效的方法。

Moreover, these neural networks could also determine the unique zoom rate that makes a singular solution appear frozen and finite.

此外，这些神经网络还可以确定使奇异解显得冻结和有限的独特缩放率。

At first, their PINN turned up only known candidates.

起初，他们的 PINN 只出现了已知的候选人。

In 2022, for instance, Buckmaster and Lai, along with Lai’s postdoctoral researcher Yongji Wang and Javier Gómez-Serrano of Brown University, used a PINN to home in on the stable blowup that Hou and Luo had found in 2013.

例如，2022 年，Buckmaster 和 Lai 以及 Lai 的博士后研究员 Yongji Wang 和布朗大学的 Javier Gómez-Serrano 使用 PINN 瞄准了 Hou 和 Luo 在 2013 年发现的稳定爆炸。

(Hou and Chen would prove its existence later that year.)

（侯和陈将在当年晚些时候证明它的存在。）

Tristan Buckmaster (left) and Javier Gómez-Serrano are using neural networks to look for extremely subtle ways of making fluid equations glitch.

Tristan Buckmaster（左）和 Javier Gómez-Serrano 正在使用神经网络寻找使流体方程出现故障的极其微妙的方法。

Dan Komoda/ Institute for Advanced Study; Jason Rossi/Brown University

Dan Komoda/高等研究院；贾森·罗西/布朗大学

They also rediscovered a known singularity candidate in the Córdoba-Córdoba-Fontelos (CCF) equations, which describe a simpler one-dimensional fluid.

他们还在 Córdoba-Córdoba-Fontelos (CCF) 方程中重新发现了一个已知的候选奇点，该方程描述了一种更简单的一维流体。

That singularity candidate was especially notable — it was unstable.

那个候选奇点尤其引人注目——它不稳定。

It had been found in 2019 because the CCF equations happen to be closely related to an even simpler fluid model that’s well understood.

它于 2019 年被发现，因为 CCF 方程恰好与一个更简单且易于理解的流体模型密切相关。

But the PINN could find this solution in a more general way, and much more precisely.

但 PINN 可以以更通用、更精确的方式找到这个解决方案。

That’s because it wasn’t a simulation in the traditional sense, stepping a fluid forward in time.

那是因为这不是传统意义上的模拟，即随着时间的推移向前推进。

Rather, it went after the frozen limit directly.

相反，它直接追求冻结限制。

“There’s no time, so you don’t care that it’s unstable,” Buckmaster said.

“没有时间，所以你不在乎它不稳定，”巴克马斯特说。

“You just try to solve the [equation] itself.”

“你只需尝试求解[方程]本身。”

The mathematicians were excited by the prospect of using their PINN to find new unstable singularity candidates.

数学家们对使用 PINN 寻找新的不稳定奇点候选者的前景感到兴奋。

They teamed up with Google DeepMind and spent the next few years fine-tuning the neural network approach to look for unstable blowups in a few different classical fluid theories.

他们与 Google DeepMind 合作，在接下来的几年里对神经网络方法进行了微调，以寻找几种不同的经典流体理论中不稳定的爆炸。

Wang, now a researcher at DeepMind, led the team in switching from off-the-shelf PINNs to bespoke neural networks tailored to fit the specific fluid equations they were trying to solve.

现在 DeepMind 的研究员 Wang 带领团队从现成的 PINN 转向定制的神经网络，以适应他们试图解决的特定流体方程。

The researchers also further tuned the structure of the PINNs to guide them toward solutions with features they knew the singularities should have.

研究人员还进一步调整了 PINN 的结构，以引导他们找到具有他们知道奇点应具有的特征的解决方案。

As they did so, their PINNs got better at spotting singularity candidates.

当他们这样做时，他们的 PINN 能够更好地发现候选奇点。

A lot better.

好多了。

These ‘unstable’ blowups have been nearly impossible to find, the ultimate needles in the haystack.

这些“不稳定”的爆炸几乎是不可能找到的，就像大海捞针一样。

In September, their collaboration of more than 20 researchers unveiled a host of singularities that had never been seen before, most of them unstable.

9 月，他们与 20 多名研究人员合作，揭示了许多以前从未见过的奇点，其中大多数都是不稳定的。

Revisiting the spinning fluid in a can, they described a collection of four new unstable singularity candidates in the Euler equations.

他们重新审视罐中的旋转流体，描述了欧拉方程中四个新的不稳定奇点候选者的集合。

These were still broadly similar to Hou and Luo’s known stable singularity, though the initial spinning conditions differed slightly in intensity and other variables.

尽管初始旋转条件在强度和其他变量上略有不同，但它们仍然与侯和罗已知的稳定奇点大致相似。

Each candidate they found was more unstable than the last — disappearing even more easily when the setup was tweaked in subtle ways.

他们发现的每个候选者都比上一个更不稳定——当设置以微妙的方式调整时，更容易消失。

They also looked at equations describing how a fluid filters through an incompressible porous medium, such as soil or rock, in two dimensions.

他们还研究了描述流体如何在二维空间中过滤通过不可压缩多孔介质（例如土壤或岩石）的方程。

No one had ever found singularity candidates in this setup.

没有人在这种设置中找到奇点候选者。

They found four — one stable, three unstable.

他们发现了四个——一个稳定，三个不稳定。

All involved a similar setup that can be visualized in a thought experiment, although in reality no scientist would be able to adjust the fluid with the endless precision necessary to make the experiment a reality.

所有这些都涉及类似的设置，可以在思想实验中可视化，尽管实际上没有科学家能够以使实验成为现实所需的无限精度来调整流体。

Imagine an ant farm filled with a sand layer and a rock layer (but no ants).

想象一个蚂蚁农场，里面充满了沙层和岩石层（但没有蚂蚁）。

Now add a blob of water, wetting some of the sand.

现在添加一滴水，润湿一些沙子。

Over time, gravity pulls the water down through the sand, and the blob flattens as it drops.

随着时间的推移，重力将水从沙子中拉下来，水滴在落下时会变平。

Eventually it smacks into the rock layer, and a property related to the fluid’s density seems to blow up.

最终它撞击到岩层中，与流体密度相关的特性似乎会爆炸。

Finally, the team returned to the one-dimensional CCF equations, this time finding an even more unstable singularity than before.

最后，团队回到一维 CCF 方程，这次发现了一个比以前更不稳定的奇点。

One way of visualizing this model is to imagine an expansive puddle with two opposing currents.

可视化该模型的一种方法是想象一个具有两个相反水流的广阔水坑。

The CCF equations describe the interface between the two currents.

CCF 方程描述了两个电流之间的界面。

If you put a carefully shaped kink into this interface, it sharpens into a singular cusp.

如果你在这个界面上放置一个精心设计的扭结，它就会锐化成一个单一的尖点。

Notably, like the Navier-Stokes equations (and unlike the other two kinds of equations the researchers studied), the CCF equations describe fluids that have a dissipation property akin to viscosity.

值得注意的是，与纳维-斯托克斯方程一样（与研究人员研究的其他两种方程不同），CCF 方程描述的流体具有类似于粘度的耗散特性。

Thus, each model they studied shows that the PINN method can handle some challenging aspect of the full Navier-Stokes equations, such as higher dimensions and dissipation.

因此，他们研究的每个模型都表明 PINN 方法可以处理完整纳维-斯托克斯方程的一些具有挑战性的方面，例如更高的维度和耗散。

“We are trying to isolate the technical difficulties one by one,” Gómez-Serrano said.

“我们正在努力将技术难题一一排除，”戈麦斯-塞拉诺说。

Crucially, none of these new singularity candidates has been proved.

至关重要的是，这些新的奇点候选者都没有得到证明。

But Gómez-Serrano expects that they can be, because the PINN’s approximations are so precise.

但 Gómez-Serrano 预计它们可以实现，因为 PINN 的近似值非常精确。

And the more precise a candidate is, the easier it is to prove that it’s a true singularity.

候选者越精确，就越容易证明它是真正的奇点。

Compared to when the group first unleashed their PINN a few years ago, they’ve gotten about a billion times more accurate .

与几年前该组织首次发布 PINN 时相比，他们的准确率提高了约十亿倍。

“The precision is remarkable,” said Eva Miranda , a mathematician at the Polytechnic University of Catalonia in Spain.

“精度非常高，”西班牙加泰罗尼亚理工大学数学家伊娃·米兰达 (Eva Miranda) 说道。

“The residual errors are so small that these solutions could realistically be used as seeds for future computer-assisted proofs.”

“残余误差非常小，这些解决方案实际上可以用作未来计算机辅助证明的种子。”

Yongji Wang, a researcher at Google Deepmind, developed many of the technical innovations that made the discovery of new unstable singularity candidates possible.

谷歌 Deepmind 研究员王永吉开发了许多技术创新，使新的不稳定奇点候选者的发现成为可能。

The million-dollar question, or technically the warm-up question to the million-dollar question, is whether the DeepMind collaboration can now use their PINN machinery to find a singularity in the Euler equations — for a fluid that isn’t trapped in a can, a much harder problem.

百万美元问题，或者从技术上来说是百万美元问题的热身问题，是 DeepMind 合作现在是否可以使用他们的 PINN 机器来找到欧拉方程中的奇点——对于没有被困在罐子里的流体来说，这是一个更难的问题。

The mathematicians say they will need to upgrade their techniques once again for this wilder and more complicated fluid, but they’re optimistic.

数学家们表示，对于这种更狂野、更复杂的流体，他们需要再次升级他们的技术，但他们很乐观。

“You’re building a robust tool to find things that are very hard to find,” Buckmaster said.

“你正在构建一个强大的工具来寻找很难找到的东西，”巴克马斯特说。

Other mathematicians, however, point out that past performance does not guarantee future returns, because an unbounded fluid is nothing like a bounded one.

然而，其他数学家指出，过去的表现并不能保证未来的回报，因为无界流体与有界流体完全不同。

“It’s a completely different animal,” said Diego Córdoba , a mathematician at the Institute of Mathematical Sciences in Spain and one of the Córdobas of the CCF model.

“这是一种完全不同的动物，”西班牙数学科学研究所的数学家、CCF 模型的科尔多巴之一迭戈·科尔多巴 (Diego Córdoba) 说。

(His father is the other.)

（另一个是他的父亲。）

And so the competition is heating up as researchers hunt for “boundary-free” singularities in the Euler equations and beyond.

因此，随着研究人员在欧拉方程及其他方程中寻找“无边界”奇点，竞争正在升温。

Córdoba and his collaborator, Luis Martínez-Zoroa of CUNEF University in Spain, have used pencil-and-paper techniques to discover stable singularities in a handful of different fluid setups .

科尔多巴和他的合作者、西班牙 CUNEF 大学的路易斯·马丁内斯-索罗亚 (Luis Martínez-Zoroa) 使用纸笔技术在几种不同的流体设置中发现了稳定的奇点。

They believe they’re on the verge of getting their approach to work for a boundary-free Euler fluid.

他们相信他们即将将他们的方法应用于无边界欧拉流体。

(Córdoba had fretted that the DeepMind collaboration was about to beat them to that goal, but to his relief, their PINNs aren’t yet powerful enough to crack the problem.

（Córdoba 曾担心 DeepMind 的合作将击败他们实现这一目标，但令他松了一口气的是，他们的 PINN 还没有强大到足以解决这个问题。

The solutions they’ve found, he said, “are unstable, but not that unstable.”)

他说，他们找到的解决方案“不稳定，但也不是那么不稳定。”）

Another competitor, Tarek Elgindi of the University of California, San Diego, has already had success working in boundary-free settings (with other caveats) and is intent on extending his strategy’s reach as well.

另一位竞争对手，加州大学圣地亚哥分校的塔里克·埃尔金迪 (Tarek Elgindi) 已经在无边界环境中取得了成功（还有其他警告），并且也致力于扩大其策略的影响范围。

It’s not clear which technique, if any, will reach the finish line.

目前还不清楚哪种技术（如果有的话）能够到达终点线。

“I’ll be very proud and very happy if Javi manages to do it,” said Córdoba, who was Gómez-Serrano’s doctoral adviser.

“如果哈维能够做到这一点，我将感到非常自豪和高兴，”戈麦斯-塞拉诺的博士生导师科尔多巴说道。

“But I’ll be even happier if we manage to do it.”

“但如果我们能够做到这一点，我会更高兴。”

If someone can, then it will be on to Navier-Stokes.

如果有人可以，那么就会交给纳维-斯托克斯。

But despite the recent rash of progress in finding new fluid glitches, mathematicians hesitate to raise their hopes too high.

然而，尽管最近在寻找新的流体故障方面取得了巨大进展，数学家们还是不愿把希望寄得太高。

“You may daydream, but only for a day or two,” Gómez-Serrano said.

“你可能会做白日梦，但只能持续一两天，”戈麦斯-塞拉诺说。

“You don’t have good enough ideas.

“你的想法还不够好。

Then the daydream stops.”

然后白日梦就停止了。”

Clarification: January 13, 2026 This article has been updated to reflect Ching-Yao Lai’s contribution to the development of the PINN method for finding blowup candidates.

澄清：2026 年 1 月 13 日 本文已更新，以反映 Ching-Yao Lai 对开发用于寻找爆炸候选者的 PINN 方法的贡献。

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