AI Finds Hidden Glitches in the Navier-Stokes Equations: Unstable Blowups and the Search for Fluid Singularities

The Navier-Stokes equations, central to modeling fluid motion, can admit mathematical solutions that defy physical interpretation. This episode explains how researchers examine when these solutions blow up, becoming infinite, and how computer simulations—despite their pixel limits—reveal subtle instabilities. It also highlights a breakthrough: artificial intelligence, in particular physically informed neural networks, helping researchers identify unstable blowups in more complex fluid setups. The hunt for a full understanding of these blowups underpins the Clay Institute prize for Navier-Stokes, and it showcases a vibrant, decades-spanning math-physics collaboration on the frontiers of fluid dynamics.

Overview

Quanta’s discussion centers on the Navier-Stokes equations, the fluid-dynamics counterpart to Newton's second law, and how mathematicians treat their solutions as mathematical objects that can, in principle, do anything, including becoming undefined or infinite. The episode situates NS as powerful for modeling real-world flows while highlighting the ongoing math problem of when these equations yield physically meaningful results across all times and places.

"the Navier-Stokes fluids almost don't want to blow up" is a helpful anthropomorphism used to describe how viscosity and energy dissipation typically tamp down extreme behaviors, yet the mathematical landscape can still harbor glitches known as blowups or singularities. The host and Charlie Wood unpack how scientists use two complementary tools: careful numerical simulations and rigorous mathematical analysis to probe possible blowups.

One of the central distinctions introduced is between stable and unstable singularities. A stable singularity is resilient to small perturbations, including the inevitable pixelation of a computer simulation; an unstable singularity, by contrast, is exquisitely sensitive to tiny changes and cannot be reliably detected by frame-by-frame simulations. This distinction underpins the search strategy and motivates the use of AI as a second lens for discovery.

***"unstable singularities can kind of spring up out of nowhere" - Quanta

From Equations to Simulations

The podcast explains how the NS equations describe fluids by applying force to different points in the fluid and watching what happens over time, yielding a vast array of possible solutions. Because fluids are distributed in space and time, researchers can’t rely on simple closed-form functions; they instead use digital simulations to advance the system forward in tiny steps. Yet this frame-by-frame approach, while essential, introduces artifacts, because a computer cannot represent infinity and must approximate derivatives and continuity. These digital artifacts complicate the interpretation of an apparent blowup in a simulation, making careful mathematical insight essential.

***"a physically informed neural network to solve differential equations" - Buckmaster

Stable vs Unstable Singularities: Why It Matters

The discussion distinguishes stable from unstable blowups. A stable blowup persists under small perturbations, suggesting a robust mathematical phenomenon that could, in principle, be provable under certain simplified conditions. An unstable blowup, however, is extremely sensitive to tiny changes and may not be detectable by conventional simulations. This sensitivity is a major reason why some researchers believe that if blowups exist in the full NS equations, they are likely unstable.

***"unstable singularities can kind of spring up out of nowhere" - Quanta

Pioneering Stable Blowups in Simplified Systems

Researchers pursue stable blowups by restricting the problem to simpler geometries or removing viscosity. The most famous candidate appeared in 2013 when Thomas Ho and Go Lo analyzed a spinning cylinder using the Euler equations (which lack viscosity). Over the following decade, Ho and his student Jia Jie Chen (now at the University of Chicago) provided rigorous proofs under these constrained conditions that the blowup is real, strengthening the case that stable blowups can exist in simplified models.

***"the most famous one and kind of the landmarks or frontier of this field, the candidate was found in 2013 by Thomas Ho" - Charlie Wood

AI-Driven Discoveries: Unstable Blowups in More Complex Setups

Beyond the stabilized, reduced models, researchers have turned to artificial intelligence to explore unstable blowups. Buckmaster, Yi, Wang, and Gomez Serrano have developed physically informed neural networks that solve differential equations by fitting the entire solution structure directly, rather than stepping forward in time. This approach can sidestep error accumulation that plagues time-evolving simulations, enabling the discovery of unstable blowups in three fluid setups, including variations on the cylindrical-can configuration used in earlier work.

***"last fall, they unveiled 5 or 10 new unstable candidate blowups in 3 fluid setups" - Charlie Wood

The Road Ahead: From 2D, 3D, to Boundary-Free Euler

The next major frontier highlighted is the boundary-free Euler equations, spinning cylinders without viscosity and in a truly three-dimensional, unbounded domain. This is viewed as a critical stepping stone toward addressing the full Navier–Stokes problem. The NSF/Clay Prize—one of the most coveted in mathematics—remains unclaimed: proving a stable blowup exists or proving none exists would win a prize worth one million dollars. The field is thus a mosaic of pencil-and-paper insights, intelligent simulations, and AI-driven methods, all progressing toward a shared goal: understanding whether NS can admit a singularity in 3D with viscosity and no boundaries.

***"If you can prove that one exists or prove that none exists, there's a million dollars price for that." - Charlie Wood

Conclusion: The Love of Pure Truth in a Concrete Problem

The episode closes by celebrating the deep, almost religious passion in pure mathematics for the truth of the subject, even when the problem is as abstract and difficult as proving blowups in NS. The conversation reflects the sense that the fluid equations sit at a striking intersection of physics, geometry, computation, and AI, drawing enthusiasts to a challenging, decades-spanning quest for fundamental understanding.

***"This really is the abstract love of mathematical truth, just for the love of the game" - Charlie Wood