内容来源：https://www.quantamagazine.org/using-ai-mathematicians-find-hidden-glitches-in-fluid-equations-20260109/

内容总结：

数学家用人工智能在流体方程中发现隐藏的“奇点”

近两个世纪以来，描述流体运动的纳维-斯托克斯方程一直是流体力学理论的基石。然而，数学界长期怀疑，在某些极端条件下，该方程可能失效，出现物理上无法解释的“奇点”——例如预测流体速度无限增大。证明奇点的存在与否，是悬赏百万美元的数学难题。

以往研究多在简化模型中发现“稳定奇点”，即容易通过多种条件触发的奇点。但在更接近真实流体的模型中，奇点（如果存在）被认为是“不稳定”的，仅会在极其精密的初始条件下出现，如同“大海捞针”，难以寻觅。

近期，一项由纽约大学、布朗大学和谷歌DeepMind等机构超过20名研究人员合作的研究取得突破。他们利用一种“物理信息神经网络”（PINN）作为新型探测工具，成功在多个经典流体方程中找到了此前未知的奇点候选对象，其中大部分为不稳定的奇点。

研究团队首先在描述三维无粘性流体的欧拉方程中，发现了四个新的不稳定奇点候选。随后，在描述流体在不可压缩多孔介质（如土壤）中二维渗流的方程中，首次找到了四个奇点候选（一个稳定，三个不稳定）。此外，他们还在一个一维流体模型中，发现了一个比以往所知更不稳定的奇点。

这些发现尚未被严格证明为真正的奇点，但PINN提供的近似解精度极高，误差极小，为后续的计算机辅助证明提供了强有力的“种子”。西班牙加泰罗尼亚理工大学数学家埃娃·米兰达评价称，其精度“非常出色”。

当前，多个研究团队正竞相寻找更困难、无边界约束条件下的流体奇点。尽管DeepMind团队的新方法展示了强大潜力，但能否最终攻克欧拉方程乃至纳维-斯托克斯方程的奇点难题，仍是未知数。正如研究人员所言，可以“做一两天白日梦”，但前路依然需要更多突破性的想法。这场利用人工智能辅助、探寻流体方程深层奥秘的竞赛，正在加速进行。

中文翻译：

借助人工智能，数学家发现流体方程中的隐藏缺陷

引言
近两百年前，物理学家克劳德-路易·纳维和乔治·加布里埃尔·斯托克斯最终完善了描述流体涡旋运动的一组方程。近两个世纪以来，纳维-斯托克斯方程始终是描述现实世界流体行为的权威理论——从穿行于大陆间的洋流，到包裹飞机机翼的气流，无不适用。

然而，许多数学家怀疑这些方程深处潜藏着缺陷。他们直觉认为，在某些情况下，该理论会失效。此时，方程将预测出违背物理规律、难以理解的流体运动方式——例如，流体可能旋转成速度超乎想象的涡旋，或是瞬间逆流。正如数学家所言，方程中的某些量将无限增大，发生"爆破"。

尽管付出巨大努力，至今无人能构建出纳维-斯托克斯方程失效的场景。若能实现这一突破（或反之证明方程永不爆破），将获得百万美元奖金。因此，作为攻克纳维-斯托克斯问题的前奏，数学家们已在各类简化流体方程中搜寻爆破现象（亦称奇点），例如仅适用于一维空间的方程。

他们确实有所发现。但本质上，所有已识别的奇点都是"稳定的"，这意味着它们能以多种方式形成。而在包括纳维-斯托克斯方程在内的最接近现实的流体理论中，爆破现象（如果存在）可能极为微妙，需要以难以想象的精确方式才会出现。这类"不稳定"的爆破如同草堆里的终极细针，几乎不可能被找到。

普林斯顿大学数学家查理·费弗曼表示："许多学者认为这些理论中存在奇点，但它们极不稳定，因此我们永远无法观测到。"正是他提出了价值百万美元的纳维-斯托克斯难题。

如今，一个数学家团队开发出训练机器识别这些幽灵般缺陷的方法。在9月发布的预印本论文中，他们重新研究了已知存在稳定奇点的简化流体方程，并发现了更多潜在的爆破场景——包括不稳定的情况。这是在多维流体中首次发现可能的不稳定奇点。

该团队继续在其他几种流体方程中发现了一系列不稳定奇点候选对象。虽然尚未找到价值百万美元的奇点，且仍需严格证明已发现的候选对象确实会发生爆破，但他们在简单模型中成功揭示潜在不稳定奇点的突破，为在更复杂场景中发现不稳定爆破带来了希望。

未参与新研究的费弗曼评价道："不稳定奇点的特性不再构成发现奇点的障碍。"

奇点猎寻
纳维-斯托克斯方程的解能捕捉永恒的一瞬。根据流体初始状态求解方程，可以推知流体在空间各点、时间各刻的速度分布。在简单解中，流体可能始于平静并永远保持静止；在复杂设定下，温和的水流可能汇成漩涡与湍流。核心谜题在于：每个满足纳维-斯托克斯方程的解——每一种可能的流体演化历程——是否在任何时空都保持物理合理性。

由于处理三维流体纳维-斯托克斯方程的难度超乎想象，数学家们从简化版本入手。例如，欧拉方程假设流体流动时无内摩擦（即无粘性）。在这类无摩擦流体中能量不会耗散，因此比粘性流体更容易发生爆破。

但即便在此简化场景中，寻找爆破解仍非易事。流体方程通常过于复杂，难以用纸笔直接求解。常见方法是借助计算机模拟流体运动，近似把握可能引发爆破的条件。若能精确界定产生爆破的条件，便可能据此严格证明爆破真实存在。

这正是托马斯·侯和罗果在2013年采用的研究路径。他们模拟了罐装数字液体：让上半部分液体朝一个方向旋转，下半部分反向旋转，随后用欧拉方程推演其演化。最终，在罐壁边界处反向流交汇的位置，涡度（衡量液体绕点旋转程度的指标）急剧增大，甚至超出计算机处理能力。

这暗示类似条件可能导致爆破，但并非确证。费弗曼指出："三维欧拉方程疑似奇解的墓地早已遍布残骸。"

侯与另一位合作者陈佳杰耗费近十年才摘除"疑似"标签。2022年，他们通过计算机证明该候选奇点暗示着真实奇点的存在。这项里程碑式的证明令数学家们渴望将前沿推向更深处。

该研究依赖计算机模拟，意味着对数字流体初始状态的微小调整（或任何数字舍入误差）不会改变流体的最终命运。即使过程略有差异，罐壁边界处仍会出现奇点。

因此，该奇点是稳定的。但奇点未必稳定。爆破可能仅在最精密的流体初始配置下发生。这种情况下，对初始设置的任何微小调整都将阻止流体爆破。

许多数学家推测，若更符合现实的流体方程中潜藏奇点，它们将呈现此类不稳定性，毫无征兆地突然出现。

这样的奇点也将更难被发现。

有限化求解
纽约大学数学家特里斯坦·巴克马斯特指出，用计算机模拟追踪不稳定奇点候选对象基本不可能实现。首先需要宇宙级的运气才能恰好设定正确的流体初始配置——这好比尝试将钢笔精确平衡于笔尖。其次，为保持平衡，还需完美无瑕地推演流体每时每刻的演化，因为最微小的偏差都会使其偏离爆破路径。

计算机无法实现无限精度，必然会引入数值误差。这些误差虽微小，却足以阻止不稳定奇点形成。巴克马斯特比喻道："就像风吹动了你的钢笔。"

因此，几乎所有已发现的爆破候选对象都是稳定的。

于是巴克马斯特与同事开始研究可能"防风"的寻找不稳定奇点之法。

这并非他们最初的计划。2021年，他们运用神经网络作为新工具，无差别搜寻各类奇点候选对象。神经网络本质上是由大量数值定义的函数，通过高效"训练"流程（包括猜测、检验和优化）精细调整这些数值，直至函数能执行特定任务。例如，用数千张标注过的猫狗照片校准神经网络后，它便能"学会"对从未见过的新图像进行"猫"或"狗"的分类。

巴克马斯特团队转向了物理信息神经网络。与图像分类神经网络不同，PINN不通过外部数据学习，而是接收描述系统随时间变化的偏微分方程，通过自我调整最终生成能求解该方程的函数。例如，它可以接收流体方程，通过训练逼近能描述流体有效演化史的函数——其中可能包含奇点。

但任何计算机技术都无法直接呈现奇点的无限特性。想象运行流体模拟并观察其随时间演化：可将流体中不同点的速度等量表示为图表曲线。随着流体变化，曲线也会如电影画面般改变。若曲线在连续帧间急剧变陡，可能预示流体正逼近奇点。然而模拟无法抵达最终终点——在曲线趋于无限陡峭前，计算机内存就会耗尽导致程序崩溃，从而无法确知是否真正走向爆破。

为规避无限性难题，数学家近期将搜索焦点转向具有自相似特性的奇点。这意味着存在一种拉伸方法，能使较早帧的速度曲线与较晚帧的更陡曲线相匹配。因此，要捕捉潜在奇点，无需再试图观察曲线无限变陡的过程，而是可以在电影播放时放大曲线的陡峭段，同时采用抵消陡峭化的缩放方式。从这个动态新视角观察，曲线将无限逼近一条具有有限陡度的冻结曲线。这种变换使目标——冻结极限——成为有限计算机能够处理的有限对象。

巴克马斯特团队意识到，PINN可能是寻找流体方程冻结解的极高效方法。更重要的是，这些神经网络还能确定使奇解呈现冻结有限状态所需的独特缩放速率。

最初，他们的PINN仅发现了已知候选对象。例如2022年，巴克马斯特与布朗大学哈维尔·戈麦斯-塞拉诺及合作者运用PINN定位了侯和罗在2013年发现的稳定爆破（侯和陈在同年晚些时候证明了其存在性）。

他们还在描述一维简化流体的科尔多瓦-科尔多瓦-丰特洛斯方程中重新发现了已知奇点候选对象。该候选对象尤为引人注目——它是不稳定的。由于CCF方程恰好与更简单易懂的流体模型密切相关，该奇点于2019年被发现。但PINN能以更普适的方式、更高精度找到这个解，因为它并非传统意义上逐步推进流体演化的模拟，而是直接追寻冻结极限。

"这里不存在时间维度，因此不稳定性无关紧要，"巴克马斯特解释道，"你只需尝试求解方程本身。"

不稳定性的新发现
巴克马斯特和戈麦斯-塞拉诺对运用PINN寻找新的不稳定奇点候选对象充满期待。他们与谷歌DeepMind合作，历时数年优化神经网络方法，在几种经典流体理论中搜寻不稳定爆破。现任DeepMind研究员的王永吉领导团队，将通用PINN升级为针对特定流体方程定制的专用神经网络。研究者还进一步调整PINN结构，引导其寻找具有奇点应有特征的解。

随着改进深入，他们的PINN识别奇点候选对象的能力显著提升——取得了质的飞跃。

今年9月，这个超过20名研究者的合作团队公布了大量前所未见的新奇点，其中多数为不稳定奇点。

重新审视罐中旋转流体时，他们在欧拉方程中描述了四个新的不稳定奇点候选对象。这些候选对象与侯和罗已知的稳定奇点大体相似，但初始旋转条件在强度等变量上略有差异。每个新发现的候选对象都比前一个更不稳定——当设置发生微妙调整时，它们消失得更为迅速。

他们还研究了描述不可压缩多孔介质（如土壤或岩石）中二维流体渗滤的方程。此前从未有人在该设定中发现奇点候选对象，而他们找到了四个——一个稳定，三个不稳定。所有这些都涉及可通过思想实验可视化的类似设定，尽管现实中科学家无法以实验所需的无限精度调整流体。想象一个装满沙层和岩层（无蚂蚁）的蚂蚁农场，注入一团浸湿部分沙层的水。随时间推移，重力使水穿过沙层下渗，水团在下落过程中逐渐扁平，最终撞击岩层时，与流体密度相关的某个属性似乎发生爆破。

最后，团队重返一维CCF方程，这次发现了比以往更不稳定的奇点。该模型可形象理解为：想象一片广阔水洼中存在两股相向水流，CCF方程描述了两股水流间的界面。若在该界面精心设置扭结，它将锐化为奇异尖点。

值得注意的是，与纳维-斯托克斯方程相似（而不同于团队研究的另两类方程），CCF方程描述的流体具有类似粘性的耗散特性。因此，他们研究的每个模型都证明：PINN方法能处理完整纳维-斯托克斯方程的某些挑战性方面，例如更高维度和耗散效应。

"我们正尝试逐个攻克技术难点，"戈麦斯-塞拉诺表示。

关键的是，这些新发现的奇点候选对象均未获证明。但戈麦斯-塞拉诺预期它们可被证明，因为PINN的近似精度极高。候选对象越精确，证明其为真实奇点的难度就越低。相比几年前首次使用PINN时，当前精度已提高约十亿倍。

"这种精度令人惊叹，"西班牙加泰罗尼亚理工大学数学家伊娃·米兰达评价道，"残差误差极小，这些解实际上可作为未来计算机辅助证明的种子。"

突破边界的竞赛
价值百万美元的核心问题（严格说是其前置问题）在于：DeepMind合作团队能否运用PINN机制在欧拉方程中发现非受限流体的奇点——这比罐中流体问题困难得多。数学家表示需要再次升级技术以应对更复杂多变的流体，但他们持乐观态度。

"我们正在构建寻找极难发现之物的强大工具，"巴克马斯特说。

然而其他数学家指出，既往成果不能保证未来突破，因为无界流体与有界流体截然不同。"这完全是另一回事，"西班牙数学科学研究所数学家、CCF模型创始人之一迭戈·科尔多瓦表示（另一位创始人是其父亲）。

随着研究者在欧拉方程及其他方程中搜寻"无边界"奇点，竞争日趋激烈。科尔多瓦与西班牙CUNEF大学合作者路易斯·马丁内斯-索罗亚已通过纸笔演算，在若干不同流体设定中发现稳定奇点。他们相信自己的方法即将适用于无边界欧拉流体（科尔多瓦曾担心DeepMind团队会抢先突破，但欣慰地发现他们的PINN尚不足以解决该问题。他指出已发现的解"虽不稳定，但未达到那种极端程度"）。

另一位竞争者——加州大学圣地亚哥分校的塔雷克·埃尔金迪已在无边界设定中取得成果（存在其他限制条件），并致力于拓展其策略的适用范围。

目前尚不清楚哪种技术（如果存在）将最终胜出。"如果哈维能成功，我会非常自豪和高兴，"曾担任戈麦斯-塞拉诺博士导师的科尔多瓦说，"但如果我们自己能成功，我会更高兴。"

若有人实现突破，下一步将剑指纳维-斯托克斯方程。尽管近期在发现流体缺陷方面进展迅猛，数学家们仍不敢过于乐观。

"你可以做白日梦，但只能做一两天，"戈麦斯-塞拉诺坦言，"当缺乏足够好的想法时，梦就该醒了。"

英文来源：

Using AI, Mathematicians Find Hidden Glitches in Fluid Equations
Introduction
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. 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.
Nevertheless, many mathematicians suspect that glitches hide deep within the equations. They have a hunch that in certain situations, the theory fails. 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. Some quantity in the equations will grow infinitely large, or “blow up,” as mathematicians put it.
Despite immense effort, no one has been able to come up with a situation where the Navier-Stokes equations falter. Doing so — or, alternatively, proving that the equations never blow up — would come with a $1 million reward. 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.
They’ve found them. But essentially all the singularities they’ve identified have been “stable,” meaning that they can form in many possible ways. 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. These “unstable” blowups have been nearly impossible to find, the ultimate needles in the haystack.
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.
Now one group of mathematicians has developed a way of training machines to spot these phantom glitches. In a preprint posted in September, they reexamined simpler fluid equations already known to host a stable singularity. There, they found additional potential blowup scenarios — including unstable ones. It was the first time a possible unstable singularity was uncovered in a fluid of more than one dimension.
The team went on to discover an assortment of unstable singularity candidates in several other fluid equations as well. They have not found any million-dollar singularities. And they still need to rigorously prove that the ones they have found do indeed blow up. 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.
“The idea of an unstable singularity no longer prevents the discovery of the singularity,” said Fefferman, who was not involved in the new research.
Singularity Hunting
A solution to the Navier-Stokes equations captures a slice of eternity. 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. In one simple solution, a fluid might start calm and remain calm forever. In a more complicated setup, gentle currents might merge into whirlpools and eddies. The great mystery is whether every solution — every single possible fluid history that satisfies the Navier-Stokes equations — makes sense everywhere and always.
But tackling the Navier-Stokes equations for fluids in three dimensions is unspeakably difficult, so mathematicians have started with easier versions of the problem. For instance, the Euler equations assume that fluids flow with no internal friction, or viscosity. Energy doesn’t dissipate in these frictionless fluids, so they should blow up more easily than viscous ones.
But even in this simpler scenario, finding a blowup solution is hard. Fluid equations are generally too complicated to solve directly with pencil and paper. 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. 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.
That’s the approach that Thomas Hou and Guo Luo took in 2013, when they simulated a digital liquid in a can. 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. 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.
This was a hint that a similar set of conditions would lead to a blowup. But it was not a guarantee. “The graveyards are strewn with alleged singular solutions of 3D Euler,” Fefferman said.
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. It was a landmark proof, and it got mathematicians hungry to push the frontier even further.
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. A singularity would still occur at the can’s boundary even if things played out a little bit differently.
Because of this, the singularity was stable. But a singularity need not be stable. A blowup might occur only when the fluid is set up in the most delicate of ways. In such a case, any adjustment to that initial arrangement, no matter how small, would prevent the fluid from blowing up.
Many mathematicians conjecture that if singularities do lurk in more realistic fluid equations, they’ll be unstable like this, springing up without warning.
They’ll also be far harder to find.
Going Finite
It’s essentially impossible to track down an unstable singularity candidate with a computer simulation. 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. 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.
Computers aren’t capable of infinite precision. They’ll inevitably introduce numerical errors that, though tiny, will stop the unstable singularity from forming. “It’s like the wind blowing on your pen,” Buckmaster said.
As a result, almost all blowup candidates have been stable.
So Buckmaster and his colleagues began to work out a potentially wind-proof way of finding unstable ones.
They didn’t set out to do so. In 2021, they used a neural network as a new way to search indiscriminately for singularity candidates of any kind. A neural network is, in general, a function defined by a vast array of numbers. These numbers get carefully adjusted through a highly efficient “training” process of guessing, checking, and refining until the function can perform some desired task. 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.”
Buckmaster and his team turned to what’s known as a physics-informed neural network, or PINN. Unlike an image-classifying neural network, a PINN doesn’t learn by studying external data. 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. 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.
But no computer technique can directly render the infinite nature of a singularity. Imagine playing the simulation of your fluid and watching it move forward in time. You might represent some quantity, such as the velocity at different points in the fluid, as a curve on a graph. As the fluid changes over time, you’ll see that curve change as well, like a movie. If the curve gets much steeper from one frame to the next, the fluid might be approaching a singularity. The simulation, however, can’t reach that final destination. The computer will run out of memory before the curve gets infinitely steep, crashing the program. Then you can’t know for sure what was going to happen — if you were truly headed for a blowup or not.
To sidestep the inconvenience of infinity, mathematicians have recently focused their search on singularities with a special property called self-similarity. 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. And so, if you want to catch a potential singularity, you no longer need to try to watch the curve get infinitely steep. Instead, you can zoom in on the steepening section of the curve while the movie plays in a way that neutralizes the steepening. From this new, dynamic perspective, the curve gets closer and closer to a frozen curve of finite steepness instead. This transformation renders the target — the frozen limit — a finite object that a finite computer can handle.
Buckmaster’s team realized that PINNs could be an extremely efficient way of finding these frozen solutions to fluid equations. Moreover, these neural networks could also determine the unique zoom rate that makes a singular solution appear frozen and finite.
At first, their PINN turned up only known candidates. In 2022, for instance, Buckmaster, Javier Gómez-Serrano of Brown University, and their collaborators used a PINN to home in on the stable blowup that Hou and Luo had found in 2013. (Hou and Chen would prove its existence later that year.)
They also rediscovered a known singularity candidate in the Córdoba-Córdoba-Fontelos (CCF) equations, which describe a simpler one-dimensional fluid. That singularity candidate was especially notable — it was unstable. 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. But the PINN could find this solution in a more general way, and much more precisely. That’s because it wasn’t a simulation in the traditional sense, stepping a fluid forward in time. Rather, it went after the frozen limit directly.
“There’s no time, so you don’t care that it’s unstable,” Buckmaster said. “You just try to solve the [equation] itself.”
A New Stable of Instability
Buckmaster and Gómez-Serrano were excited by the prospect of using their PINN to find new unstable singularity candidates. 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. Yongji 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. The researchers also further tuned the structure of the PINNs to guide them toward solutions with features they knew the singularities should have.
As they did so, their PINNs got better at spotting singularity candidates. A lot better.
In September, their collaboration of more than 20 researchers unveiled a host of singularities that had never been seen before, most of them unstable.
Revisiting the spinning fluid in a can, they described a collection of four new unstable singularity candidates in the Euler equations. 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. Each candidate they found was more unstable than the last — disappearing even more easily when the setup was tweaked in subtle ways.
They also looked at equations describing how a fluid filters through an incompressible porous medium, such as soil or rock, in two dimensions. No one had ever found singularity candidates in this setup. They found four — one stable, three unstable. 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. Imagine an ant farm filled with a sand layer and a rock layer (but no ants). Now add a blob of water, wetting some of the sand. Over time, gravity pulls the water down through the sand, and the blob flattens as it drops. Eventually it smacks into the rock layer, and a property related to the fluid’s density seems to blow up.
Finally, the team returned to the one-dimensional CCF equations, this time finding an even more unstable singularity than before. One way of visualizing this model is to imagine an expansive puddle with two opposing currents. The CCF equations describe the interface between the two currents. If you put a carefully shaped kink into this interface, it sharpens into a singular cusp.
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. 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.
“We are trying to isolate the technical difficulties one by one,” Gómez-Serrano said.
Crucially, none of these new singularity candidates has been proved. But Gómez-Serrano expects that they can be, because the PINN’s approximations are so precise. And the more precise a candidate is, the easier it is to prove that it’s a true singularity. Compared to when the group first unleashed their PINN a few years ago, they’ve gotten about a billion times more accurate.
“The precision is remarkable,” said Eva Miranda, a mathematician at the Polytechnic University of Catalonia in Spain. “The residual errors are so small that these solutions could realistically be used as seeds for future computer-assisted proofs.”
Race To Escape the Boundary
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. The mathematicians say they will need to upgrade their techniques once again for this wilder and more complicated fluid, but they’re optimistic.
“You’re building a robust tool to find things that are very hard to find,” Buckmaster said.
Other mathematicians, however, point out that past performance does not guarantee future returns, because an unbounded fluid is nothing like a bounded one. “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. (His father is the other.)
And so the competition is heating up as researchers hunt for “boundary-free” singularities in the Euler equations and beyond. 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. They believe they’re on the verge of getting their approach to work for a boundary-free Euler fluid. (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. The solutions they’ve found, he said, “are unstable, but not that unstable.”)
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.
It’s not clear which technique, if any, will reach the finish line. “I’ll be very proud and very happy if Javi manages to do it,” said Córdoba, who was Gómez-Serrano’s doctoral adviser. “But I’ll be even happier if we manage to do it.”
If someone can, then it will be on to Navier-Stokes. But despite the recent rash of progress in finding new fluid glitches, mathematicians hesitate to raise their hopes too high.
“You may daydream, but only for a day or two,” Gómez-Serrano said. “You don’t have good enough ideas. Then the daydream stops.”