For nearly two centuries, the equations governing fluid motion have been among the most trusted pillars of physics and engineering. From predicting weather systems and ocean currents to designing aircraft and modeling blood flow in the human body, the Navier–Stokes equations have long been treated as mathematically sound and physically complete.
Yet behind this apparent certainty lies one of the greatest unresolved problems in modern mathematics: Do these equations always make sense? Or do they secretly fail under extreme conditions?
Now, aided by specially trained artificial intelligence systems, mathematicians are uncovering new evidence that the equations describing fluid flow may harbor subtle and elusive breakdowns—so-called singularities—that have remained hidden despite decades of intense scrutiny.
The $1 Million Question at the Heart of Fluid Dynamics
The Navier–Stokes equations describe how fluids move through space and time. They account for velocity, pressure, viscosity, and external forces, capturing everything from gentle breezes to raging hurricanes.
But mathematicians do not yet know whether these equations always produce physically meaningful solutions. Under certain initial conditions, quantities like velocity or vorticity might grow without bound, reaching infinite values in finite time. Such behavior would be physically impossible—and mathematically catastrophic.
Proving whether such “blowups” can occur, or proving they never occur, is so important that the Clay Mathematics Institute has placed a $1 million prize on the problem.
Despite enormous effort, no one has yet claimed it.
Why Singularities Matter More Than You Think
In mathematical physics, a singularity is not merely a curiosity—it is a signal that a theory may be incomplete or internally inconsistent.
If singularities exist in fluid equations, they could imply:
- Fundamental limits to predictability in fluid motion
- Breakdowns in computer simulations used across science and engineering
- Gaps in our understanding of turbulence, chaos, and energy transfer
Conversely, proving that singularities never occur would cement the Navier–Stokes equations as one of the most robust physical theories ever formulated.
Either outcome would reshape mathematics and physics.
Stable vs. Unstable Singularities: The Real Challenge
Over the years, mathematicians have identified singularities in simplified versions of fluid equations—often in one dimension or in idealized settings. Almost all of these are stable singularities, meaning they occur even if the system’s initial conditions are slightly altered.
But experts believe that if singularities exist in realistic, three-dimensional fluids, they are likely unstable.
An unstable singularity occurs only when the system is prepared in an extraordinarily precise way. Any tiny deviation—numerical error, rounding, noise—prevents it from forming.
This makes unstable singularities nearly impossible to find using traditional simulations.
Why Computers Alone Aren’t Enough
Standard computer simulations evolve fluids forward in time step by step. But computers cannot represent infinite quantities, and they cannot maintain infinite precision.
As a fluid approaches a potential singularity, numerical errors accumulate. The simulation crashes before revealing whether the system truly blows up or merely appears to do so temporarily.
For unstable singularities, even microscopic errors completely derail the process.
As mathematician Tristan Buckmaster explains, searching for unstable singularities with traditional simulations is like trying to balance a pen perfectly on its tip while the wind is blowing.
Enter Artificial Intelligence: A New Mathematical Tool
To overcome these limitations, mathematicians turned to physics-informed neural networks (PINNs)—a specialized form of AI designed not to analyze data, but to solve equations directly.
Unlike traditional simulations, PINNs do not step through time incrementally. Instead, they search for entire solutions to partial differential equations all at once, guided by the mathematical structure of the equations themselves.
This makes them uniquely suited to finding delicate, unstable solutions that would collapse under conventional numerical methods.
Freezing Infinity: The Trick That Makes Singularities Visible
One of the most powerful ideas enabling this breakthrough is self-similarity.
In many singularities, the system looks the same at different scales as it approaches blowup. By applying a mathematical transformation that continuously “zooms in” on the evolving solution, researchers can convert an infinite process into a finite, static one.
In this transformed frame, a singularity becomes a fixed object—a frozen mathematical structure that a neural network can actually represent.
PINNs excel at finding these frozen limits.
Early Successes and a Turning Point
Initially, the AI rediscovered known singularities, validating the approach. But as researchers refined the networks—customizing architectures, optimizing loss functions, and embedding deeper physical constraints—the results became more surprising.
In 2022, PINNs successfully reproduced a landmark singularity discovered by Thomas Hou and Guo Luo in frictionless fluids.
More importantly, they went further.
The Breakthrough: Discovering New Unstable Singularities
In a major preprint released in late 2025, a collaboration involving more than 20 researchers—including scientists from Google DeepMind—announced the discovery of multiple previously unknown unstable singularity candidates across several fluid models.
For the first time, mathematicians identified unstable singularities in fluid equations involving more than one spatial dimension.
This had never been done before.
What the AI Actually Found
The team uncovered:
- Multiple unstable singularities in the Euler equations for fluids confined by boundaries
- New singularities in models of fluid flow through porous media
- Even more unstable solutions in one-dimensional dissipative fluid equations
Each discovery demonstrated that AI could navigate mathematical landscapes previously considered inaccessible.
Why This Matters for Navier–Stokes
While none of these findings yet resolve the Navier–Stokes problem itself, they represent crucial stepping stones.
Each model tested isolates a specific difficulty present in Navier–Stokes equations:
- High dimensionality
- Dissipation (viscosity)
- Boundary effects
- Extreme instability
The success of PINNs across these challenges suggests the method may eventually scale to the full problem.
Precision at an Unprecedented Scale
One of the most remarkable achievements is accuracy. Compared to early attempts, modern PINNs now achieve roughly a billion-fold improvement in precision.
This matters because mathematical proof often begins with an approximate solution that is refined and validated rigorously.
Experts believe these AI-generated candidates may soon serve as seeds for fully formal proofs.
The Growing Race Among Mathematicians
The field is now intensely competitive.
Alongside AI-driven approaches, researchers using traditional analytical techniques are also making progress in boundary-free fluid models. Different methods may converge—or one may prevail.
What is clear is that the pace of discovery has accelerated dramatically.
AI as a Partner, Not a Replacement
Importantly, AI is not “solving” mathematics independently. Human insight remains essential for interpreting results, designing transformations, and proving rigor.
What AI offers is something unprecedented: the ability to explore vast, unstable mathematical spaces that were previously unreachable.
Looking Ahead: From Euler to Navier–Stokes
The ultimate goal remains unchanged: determining whether singularities exist in the Navier–Stokes equations governing real fluids.
While optimism is growing, mathematicians remain cautious.
Progress is real—but the final leap may still require ideas not yet imagined.
Conclusion: A New Era in Mathematical Discovery
The marriage of artificial intelligence and pure mathematics is reshaping what is possible. By revealing hidden glitches in fluid equations, AI has not answered one of mathematics’ greatest questions—but it has made the impossible searchable.
Whether this path leads to a million-dollar proof or a deeper mystery, one thing is certain: the rules of mathematical discovery have changed forever.
FAQs
1. What are Navier–Stokes equations?
They are equations that describe the motion of fluids like air and water.
2. What is a singularity in fluid dynamics?
A point where mathematical quantities become infinite, signaling a breakdown of the equations.
3. Why is the Navier–Stokes problem worth $1 million?
Because it is one of the hardest unsolved problems in mathematics.
4. What role does AI play in this research?
AI helps find extremely unstable solutions that traditional simulations miss.
5. What is a physics-informed neural network?
A neural network trained directly on physical equations rather than data.
6. Have these singularities been proven?
Not yet—they are highly precise candidates awaiting rigorous proof.
7. Why are unstable singularities important?
They may exist in realistic fluids but are incredibly hard to detect.
8. Does this mean Navier–Stokes is wrong?
Not yet; the question remains open.
9. Can AI solve the Navier–Stokes problem?
AI can assist, but human proofs are still required.
10. What happens if singularities are proven?
It would reshape our understanding of fluid motion and predictability.