DeepMind’s “Disproof” of Navier-Stokes: Euler Blow-Ups and Why It’s Overhyped
This text is the end result of my playing around Chat GPT’s 5 “deep research” feature. I believe, after hours of discussion about what happened lately around the Navier-Stokes equations breakthrough, I ended up with the summary being worthy of saving up in here. For anyone who is looking for explanation of the case at the level that suits me!
Background: Navier-Stokes, Euler, and the Singularity Problem
The Navier-Stokes equations describe how fluids (like air or water) move. They include effects of viscosity (internal friction). The Euler equations are basically the Navier-Stokes equations without viscosity (an idealized “perfect fluid”). A long-standing mystery in fluid dynamics is whether these equations can ever produce a singularity, meaning a smooth flow suddenly blows up to infinities (like velocity or pressure becoming infinite in finite time) [deepmind.google]. Such a blow-up would indicate the equations break down (they no longer give physically meaningful results beyond that point). This is not expected to actually happen in the real world – if the math predicts infinite velocity, it means the model has reached a limit, since in reality something else would intervene (for example, real fluids aren’t perfectly continuous; new physics or molecular effects would kick in long before infinite anything) [deepmind.googlereddit.com]. But finding a singularity mathematically is hugely important for pure math and theoretical physics: it reveals the fundamental limits of these fluid equations and could resolve the Clay Institute’s Millennium Prize Problem on Navier-Stokes (worth $1 million) [eu.36kr.comeu.36kr.com]. The prize will be won by whoever can prove either that no singularities occur (the equations stay well-behaved for all time) or give an example where a singularity does occur.
For decades, mathematicians have suspected that in 3D (three-dimensional space) the Euler and Navier-Stokes equations do allow singularities – in other words, they expect at some extreme scenario the equations will blow up (this would “disprove” the idea that Navier-Stokes is always smooth) [reddit.com]. However, finding such a scenario explicitly has been incredibly challenging. Traditional numerical simulations often suggested blow-ups but could never be fully trusted – when computing with finite precision, an “apparent” blow-up might disappear at higher resolution [quantamagazine.orgquantamagazine.org]. In fact, in some simpler or symmetric cases, researchers have found hints of blow-up (and even computer-assisted proofs in very special cases). For example, there was a known scenario for 3D Euler in an axisymmetric domain (a symmetric vortex setup) that appears to blow up, and a recent effort even rigorously verified a singularity in a related simplified system under certain symmetry [reddit.com]. But crucially, these earlier finds were what scientists call “stable” singularities – meaning if you nudge the initial setup a tiny bit, the blow-up still happens. A stable singularity is a robust solution that doesn’t require perfect initial conditions.
Here’s the catch: experts believe that for the real Navier-Stokes problem (full 3D, no special symmetries or boundaries), any singularity would have to be unstable [circularastronomy.comeu.36kr.com]. An unstable singularity is one that requires exquisitely precise initial conditions – if you perturbed the setup even infinitesimally, the singularity would vanish (the fluid would just continue evolving smoothly) [arxiv.org]. In other words, nature would almost never stumble upon these singular flows by accident. It’s like balancing a pencil perfectly on its tip: in theory it’s possible, but the slightest breeze will topple it. Stable singularities, by contrast, are like a pencil planted in a holder – even if you wiggle it a bit, it stays up. For the 3D Euler and Navier-Stokes equations in free space (no boundaries), all evidence and intuition say stable blow-ups shouldn’t exist [eu.36kr.com]. So if singularities exist at all, they’re expected to be this precarious, unstable type. The trouble is, unstable singularities are far harder to find: you have to get the initial conditions exactly right, so traditional simulation or analytical guessing methods have struggled to pin them down [circularastronomy.com].
What Did DeepMind’s Team Do?
In September 2025, a team of researchers from DeepMind (Google’s AI lab) and mathematicians from places like NYU, Brown, and Stanford published a paper claiming the first systematic discovery of entire families of these elusive unstable singularities [arxiv.org]. In essence, they used a clever AI-driven approach to find flows that do blow up under perfect conditions. The key tool was a type of neural network called a Physics-Informed Neural Network (PINN). Instead of training on big data, PINNs are designed to obey physical equations. Here’s how it works in this case: the neural network represents the candidate solution (the fluid’s velocity field, for example), and the Navier-Stokes/Euler equations are encoded in the network’s loss function. The network is then trained to minimize the “residual”, which is basically the error in satisfying the PDE (partial differential equation) everywhere in space and time [eu.36kr.comdeepmind.google]. By doing this, the network tries to output a function that exactly satisfies the fluid equations (plus conditions for a singularity scenario).
The DeepMind team also built in a lot of expert knowledge and numerical muscle into this process. They weren’t just blindly training a vanilla neural net. They “embedded mathematical insights” about how singularities might look (for example, they focused on self-similar solutions – where the flow profiles at blow-up scale in space and time in a certain way, a common form for potential singularities) and used a special high-precision optimizer (a second-order Gauss–Newton method rather than standard gradient descent) to really dial in on a solution with extreme accuracy [arxiv.org]. They also refined the training in multiple stages, each time improving the resolution and accuracy, effectively pushing the neural network to reach near machine-level precision in satisfying the equations [arxiv.org]. In fact, they report that for some solutions the error is only on the order of round-off error of the hardware – basically as accurate as double-precision arithmetic allows [arxiv.org]. This is astounding: they likened it to predicting the diameter of the Earth within a few centimeters [deepmind.google]! Such precision is not just academic bragging rights – it’s necessary if you want to be confident you’ve truly found a singularity and not a numerical fluke, and it’s good enough to then attempt a rigorous computer-assisted proof that the singularity is real [arxiv.org].
Using this AI-augmented approach, the team searched for singularity solutions in three different fluid equations [eu.36kr.comeu.36kr.com]. Two of these were somewhat “simpler” model equations that are known to be closely related to 3D Navier-Stokes/Euler behavior, and the third was the actual 3D Euler equations (with a boundary) in a specific scenario. In particular, they looked at:
The Incompressible Porous Media (IPM) equation, a model of fluid in a porous medium which has mathematical similarities to fluid equations and can develop singular behavior.
The Boussinesq equations (in 2D), which describe a fluid with temperature differences (this is like a 2D analog of certain 3D flows and has been conjectured to blow up in scenarios analogous to Euler).
The 3D Euler equation with a boundary (essentially a fluid in a domain with a wall or boundary, rather than the whole infinite space). This setup was likely chosen because having a boundary (like a solid wall) can facilitate certain vortex stretching scenarios – in fact, one of the most famous past singularity attempts (the Hou–Luo scenario) involved fluid swirling near a boundary.
For each of these equations, the team’s method found multiple new solutions that exhibit finite-time blow-up. Crucially, these were unstable singularities: after finding each candidate solution, they performed a stability analysis (by linearizing the equations around that solution) and confirmed that it has one or more “unstable modes” – meaning if you perturb the initial state along any of those modes, the solution veers off and no blow-up occurs [arxiv.org]. In other words, you have to hit the bull’s-eye to get the blow-up. This matches exactly the expectation for the real Navier-Stokes problem: no easy, stable blow-ups, only the delicate, knife-edge unstable ones [circularastronomy.comarxiv.org]. The difference is, now we know what some of those knife-edge solutions look like! Before this work, unstable singularities were like a theoretical possibility that no one had seen; now we have explicit examples (at least for those related model equations and a form of Euler). It’s the “first systematic” discovery because they didn’t just stumble on one lucky case – they developed a general recipe to find many such singular solutions across different equations [arxiv.org].
Key Findings and Insights from the Paper
One major result of the paper is precisely identifying those new singular solutions. For example, for the incompressible porous media (IPM) equation, they found several self-similar blow-up solutions. For the 3D Euler (with boundary) case, they found new blow-up solutions as well (different from the previously known symmetric scenario) [arxiv.org]. Each solution comes with a certain blow-up “rate” – essentially how some norm of the fluid (like the peak vorticity, which measures rotation) grows as it approaches the singularity time. The team discovered an interesting pattern: as they found more and more unstable singularities (some requiring more fine-tuning than others), the blow-up rates followed a surprisingly simple trend. When they plotted the blow-up speed (technically a parameter λ that characterizes how fast the singularity grows) against the “order of instability” (basically the number of independent tiny perturbations that could spoil the blow-up, i.e. how many directions in initial-condition space you need to tune) they saw the points line up in a roughly straight line [deepmind.googleeu.36kr.com]. This suggests a linear relationship: the more unstable a singularity (the more conditions you must get exactly right), the faster it blows up. This empirical law was observed clearly in at least two of the equations (IPM and the 2D Boussinesq case) [deepmind.google]. It hints at an underlying mathematical structure unifying these solutions [eu.36kr.com]. In simple terms, it’s like they uncovered a “family” of singular solutions parameterized by how unstable they are, and a neat formula ties them together. This kind of insight is valuable because it can guide theorists – it’s a clue toward the deeper theory of singularities in fluids. It even suggests there may be infinitely many such solutions (as you go to higher and higher instability orders along that line) [deepmind.google].
Another key point is the unprecedented accuracy achieved for these solutions. The paper emphasizes that previous attempts to find singularities were hampered by limited precision – an unstable singularity might look like it’s forming, but then a tiny numerical error kicks it off track at the last moment, so you can’t be sure if the real equations would have blown up or not [quantamagazine.org]. By using their PINN + Gauss-Newton approach, the authors obtained solutions with errors as low as 10^(-15) in satisfying the equations [arxiv.org]. This level of precision means that the solution they found is extremely close to an actual exact solution of the PDE. In fact, they claim this meets the standards required for a computer-assisted proof – essentially, one can take their numerical solution and, using interval arithmetic and other rigorous techniques, put a mathematical stamp on it that “yes, a true solution with a blow-up exists in the neighborhood of this numerical approximation” [arxiv.org]. In other words, the AI didn’t just output some fuzzy, heuristic answer – it produced something solid enough that mathematicians can likely prove it’s correct. This elevates the result from a purely computational curiosity to a credible mathematical discovery.
To sum up the paper’s contributions in plain language:
Multiple new blow-up scenarios found: They found new examples where fluid equations (Euler, IPM, Boussinesq) lead to infinite values in finite time. These were scenarios nobody had explicitly seen before.
All are unstable: Each of these blow-ups will only occur under perfectly fine-tuned conditions – confirming the suspected nature of singularities in the wild 3D fluid equations [arxiv.org]. Even a minuscule perturbation would prevent the singularity (the flow would remain regular).
Self-similar and characterized precisely: The solutions have a self-similar form (they “look the same” as they blow up, just zoomed in), and the team was able to measure key parameters like the blow-up rate λ to high precision [arxiv.org].
A pattern in instability: There’s a linear relationship connecting how unstable a solution is and how fast it blows up, observed in at least two different fluid models [deepmind.google]. This discovery hints at a deeper theory and suggests where to look for even more extreme solutions.
New computational methodology: Perhaps as important as the singularities themselves, the paper demonstrates a new way to use AI/neural networks as a “discovery tool” for math and physics [deepmind.google]. By embedding the physical law into a neural network and training it with almost absurd precision, they navigated the “solution space” of the equations in a way that traditional methods couldn’t. This approach can be applied to other tough PDE problems, not just fluid dynamics [deepmind.google].
Significance: What Does It Mean (and Not Mean)?
It’s hard to overstate the mathematical significance of this work. For mathematicians studying PDEs (partial differential equations) and fluid dynamics, this is a big deal. It provides concrete evidence and examples in support of the idea that the Euler equations do develop singularities in 3D (at least in a setup with a boundary), and by extension it strongly suggests the Navier-Stokes equations likely can blow up too under the right conditions. In fact, many experts have believed for a while that finite-time blow-up is the more likely outcome for the Navier-Stokes problem [reddit.com]. This work doesn’t completely solve the Millennium Prize Problem – a formal proof would still be needed – but it pushes the frontier forward. It gives a clear target for a proof: now that we have candidate solutions and an approach to verify them, one could imagine a full rigorous proof that “there exists an initial condition for 3D Euler (or even Navier-Stokes) that blows up at time T.” Achieving that would win the $1M prize and settle the problem. The DeepMind paper is a stepping stone toward that goal, and it introduces a novel toolset (AI-assisted search) that could be used in other unsolved problems in math and physics [deepmind.google].
Beyond pure math, the authors themselves (and the media coverage) have hinted at broader impacts: “a new way to leverage AI on longstanding challenges in mathematics, physics and engineering” [deepmind.googlemedium.com]. It’s true that the methodology – combining neural networks with physical equations and extreme precision – could find use in engineering or scientific computing (for example, designing better solvers or discovering new solutions in other nonlinear systems). And certainly in the academic fluid dynamics community, having a catalogue of singular solutions enhances understanding of how and why fluid equations can fail. It might inspire new theory on turbulence or extreme events (since real fluids don’t blow up to infinity, but they can exhibit very sharp spikes; understanding the idealized blow-up might give insight into the precursors of things like extreme turbulence cascades).
However, it’s important to separate hype from reality, especially regarding “engineering applications” or everyday impact. Despite some breathless headlines, this discovery doesn’t mean we’ll suddenly build better airplanes or predict hurricanes more accurately next year. Why not? Because the singularities in question are fundamentally non-physical scenarios – they require conditions so precise that nature essentially never realizes them, and even if it tried, real-world physics (like viscosity, heat conduction, molecular discreteness, etc.) would avert the literal mathematical singularity. As the DeepMind team itself points out, these blow-ups are “situations which could never physically happen”, crafted precisely to probe the limits of the theory [deepmind.google]. So, while one article excitedly claimed this breakthrough “will bring new breakthroughs to […] weather forecasting, flood simulation, aerodynamics, and even cardiovascular research” [eu.36kr.com], that’s an overstatement. Engineers and physicists dealing with real fluids don’t intentionally set up perfectly symmetric, infinitesimally tuned flows in hopes of making them explode. In fact, any tiny real-world imperfection will keep the flow regular (or cause turbulent mixing long before an “infinite” blow-up), and viscosity – however small – tends to smooth out the very sharp gradients that lead to blow-up.
So, from an engineering perspective, nothing immediate changes. For instance, you don’t have to worry that the air flowing around your airplane wing will suddenly accelerate to infinite speed – it won’t, and this research doesn’t suggest otherwise (on the contrary, it confirms that such extremes are essentially mathematical curiosities requiring unrealizable perfection). As some experts noted in discussions, if a Navier-Stokes blow-up were possible for a real fluid, it would imply something physically bizarre like a blob of honey spontaneously “exploding” into chaos despite viscosity – which we simply don’t observe [reddit.com]. Thus, these singularities mainly tell us where the equations break: if you drove a fluid in just the “right” (mathematically contrived) way, the ideal equations say “I can’t cope beyond here.” In practice, something like turbulence or molecular effects would intervene, meaning the continuum Navier-Stokes model ceases to be valid at that point [reddit.comreddit.com].
From the pure mathematics viewpoint, though, this work is exciting rather than disappointing. It essentially provides a big piece of evidence in favor of the “blow-up” side of the Navier-Stokes conjecture. It demonstrates a powerful new technique for discovering solutions that were previously invisible to us. And it shows that AI can be used in a very targeted, precision way – not just for approximate or heuristic outcomes, but to genuinely aid in mathematical discovery and proof. As one of the authors put it, by combining deep learning with math insights, they turned PINNs into a “discovery tool that finds elusive singularities” [deepmind.google]. This could herald more AI-assisted mathematical breakthroughs (DeepMind has openly expressed interest in helping solve major open problems, and here we see a concrete success).
Why Does It Feel Overhyped? (Possible Weak Links and Caveats)
The news of this achievement was accompanied by some sensational headlines – for example, calling it “a century-old math mystery cracked” or suggesting DeepMind might snag a Clay Millennium Prize or a Nobel Prize [eu.36kr.com]. It’s understandable: the Navier-Stokes problem is famous, and anything AI + big math problem makes for good press. But let’s dial back the hype and look at the caveats:
Not the final proof for Navier-Stokes: The work did not directly find a singularity in the actual 3D Navier-Stokes equations (with viscosity) in an unbounded domain, which is the strict statement of the Clay Prize problem. They tackled Euler (inviscid flow) and related models, and in Euler’s case they even added a boundary. The Clay problem is specifically about Navier-Stokes in either {R}^3 or, say, a periodic box, with nice initial data. So, we don’t yet have “Navier-Stokes disproved” in the sense of a counterexample that officially solves the problem. We do have a stronger intuition and evidence that such a counterexample is out there, and perhaps a roadmap to find it. But until a singularity is either found in the viscous case or we manage to translate the Euler singularity into Navier-Stokes, the million dollars isn’t claimed. (Many believe that if Euler blows up, Navier-Stokes will blow up too – since viscosity is like a small regularization, not an all-powerful preventive mechanism [reddit.com] – but this still needs confirmation.) In short, DeepMind’s work is a huge step, not the final step.
Reliance on numerics: At the end of the day, the singularities were found by numerical optimization (neural network training). No matter how precise, there’s always the question: could there be a hidden numerical artifact? The team mitigated this by reaching extreme precision and checking stability etc., so the results are convincing. But skeptics might wait for the promised “rigorous mathematical validation via computer-assisted proofs” to be absolutely sure [arxiv.org]. It’s one thing to say “our neural net solution didn’t diverge when we refined it, so we’re pretty sure,” and another to have a full theorem with inequalities showing a true solution exists. The good news is the latter seems within reach now, but it’s not published yet aside from the arXiv preprint.
Unstable = very fragile: By definition, these solutions are extremely fragile. If your numerical method isn’t accurate enough, you won’t find them (the team had to tune things just right). If you try to simulate these initial conditions in a normal fluid solver without the special approach, round-off error would probably spoil the blow-up. So independent reproduction is non-trivial – it requires similar high-precision methods. It’s not a trivial validation where anyone can run a code and see the fluid blow up; the blow-ups hide behind a “wall” of instability. This also underscores that they’re not physically robust. It’s a mathematical existence proof more than something you could demonstrate in a lab experiment. The “infinite precision” initial data requirement [arxiv.org] is essentially a theoretical construct.
Physical relevance: As discussed, these singularities expose the limitations of the equations but don’t directly translate to practical situations. Real fluids have boundaries, noise, viscosity, and other effects that nature uses to avoid infinities. So the direct practical impact is minimal. Some media pieces did over-hype this aspect, implying this will revolutionize fields outside of pure math. In reality, it’s mainly a victory for mathematicians and theoretical physicists. Calling it “math purist hype” is not far off – the excitement is largely within the realm of mathematics: solving (or making progress on) a famous conjecture and showcasing a new way to do math research. Engineers won’t be designing new pumps or planes based on unstable singular flows that occur only on paper.
Next steps needed: There are still weak links to firm up. For one, extending the result to the genuine Navier-Stokes equations with viscosity is an obvious challenge – the current paper didn’t explicitly report a singular solution for Navier-Stokes itself. It’s possible that a similar approach could find a blow-up in Navier-Stokes (perhaps in the limit of very low viscosity), but it might be even harder to pin down because viscosity tends to smooth out the flow a bit. The team will likely try this or encourage others to. Additionally, turning these findings into a formal proof will require careful analytical work on top of the numerics. The authors note that their solutions meet the requirements for a rigorous proof check [arxiv.org], so we can expect follow-up papers by mathematicians (possibly some of the co-authors like Buckmaster, Gómez-Serrano, etc.) to finalize the proof that “yes, Euler equations can blow up in finite time under these initial conditions.” Only after that, and ideally after showing the Navier-Stokes analog, can we truly say the Navier-Stokes millennium problem is solved. For now, we have a very promising experimental (in the computational sense) breakthrough.
In summary, the announcement might feel overhyped because phrases like “DeepMind solved Navier-Stokes” circulated, but what actually happened is more nuanced. DeepMind’s AI didn’t magically solve the equations in a general sense – it was guided by a lot of human mathematical insight and focused computational power to find specific extreme-case solutions. This is an amazing achievement, but it lives in the world of theoretical math. It confirms what many suspected (that singularities do exist in these equations) with concrete examples, rather than overturning our understanding of everyday fluids. The hype is partly because it’s a prestigious problem and AI was involved, and indeed it is a milestone worth celebrating in the math community. We just have to keep in mind where its impact is felt: mostly in advancing mathematical knowledge and techniques, rather than in practical fluid engineering.
High-Level Summary of the Paper
To put it all in a straightforward storyline:
The longstanding puzzle: Can fluid equations (Euler/Navier-Stokes) produce infinite behavior (a singularity), or do they remain smooth forever? This is unanswered and very important in math. If a singularity exists, it likely must be an “unstable” one requiring perfect setup [circularastronomy.com].
What was done: A collaboration between an AI team (DeepMind) and mathematicians used a physics-informed neural network approach to search for those elusive singular solutions. They set up a neural network to satisfy the equations and homed in on solutions that blow up, with extremely high precision [eu.36kr.comarxiv.org].
What they found: Multiple new examples of singularities in fluid equations (including the 3D Euler case) that were previously unknown. All of these are unstable singularities, as expected for the real 3D problem [arxiv.org]. They also found a simple linear pattern relating the singularity’s growth rate to how unstable it is [deepmind.google].
Why it matters: This is the first time unstable singularities have been systematically identified and characterized. It provides strong evidence and candidates for the kind of blow-up that could resolve the Navier-Stokes conjecture [arxiv.org]. It also showcases a new paradigm of using AI as a tool for discovering fundamental mathematical phenomena [deepmind.google].
Why caution is warranted: These singularities are mathematical extreme cases, not something that occurs in real-life fluid flow (they require idealized, perfectly tuned conditions) [deepmind.google]. The results, while groundbreaking in theory, don’t directly translate to technological or engineering improvements in the near term. And while the work is a big leap forward, a formal proof for Navier-Stokes isn’t in hand yet – but it’s now on the horizon.
In conclusion, DeepMind’s paper marks a major breakthrough in mathematical fluid dynamics, unearthing solutions that had been conjectured but never seen. It has somewhat “disproved” the always-smooth Navier-Stokes scenario, at least in the inviscid limit, by demonstrating how blow-ups can occur. The reason it might feel overhyped is that the practical world is unlikely to ever see these blow-ups occur – they are a triumph of mathematical insight and computational power more than a change in how we model everyday fluids. Nonetheless, for the math and physics community, it’s a significant and exciting development, and it opens the door to finally settling a 150-year-old question with the help of modern AI tools. The work exemplifies how AI and human expertise together can crack problems once thought too hard to tackle, even if the impact is mostly on theoretical understanding rather than on immediate real-world applications.
Sources: The description of singularities and their importance draws from DeepMind’s announcement blog and related explainers [deepmind.googlemedium.com]. The distinction between stable and unstable singularities is emphasized in both the paper and explanatory posts [circularastronomy.comarxiv.org]. Details of the AI approach and precision come from the authors’ arXiv paper [arxiv.org] and media coverage [eu.36kr.com]. The discovery of the new family of solutions and the linear pattern in blow-up rates are noted in the DeepMind blog and press articles [deepmind.googleeu.36kr.com]. Cautions about physical relevance and overhype are informed by the DeepMind blog itself (noting these scenarios aren’t physical) [deepmind.google] and expert commentary on the implications [reddit.comreddit.com]. Overall, the analysis reflects the high-level findings of Wang et al. (2025), “Discovery of Unstable Singularities” [arxiv.orgarxiv.org], and situates them in context.