Stable Singularity of the Euler Equations on R^3 without forcing – Anima on AI

Researchers using a physics-informed neural network (PINN) report evidence of a stable, self-similar finite-time blowup for the incompressible Euler equations on R^3 without boundary or forcing, a result that would resolve a long-standing open problem in fluid dynamics. The team, which includes Anima Anandkumar, refined the PINN solution with spline representations to obtain tight certified bounds and analyzed the transport field to support stability. The work contrasts with a separate announcement by Alpöge and Tristan Buckmaster on Euler equations with forcing, and highlights the role of physics-informed AI in mathematical discovery.

September 7, 2026 In the theory of partial differential equations PDEs , an important problem is analyzing whether there is loss of regularity in finite time, even when there is smooth initial data. This is known as singularity formation, or blowups. A famous one is whether fluid dynamics represented by Navier-Stokes equations experiences such blowups, and is one of the unsolved Millennium prize problems. A closely related problem is known as the Euler problem and it represents inviscid flow, i.e., it lacks the viscosity term present in the Navier-Stokes equation. Intuitively the lack of viscosity makes it easier for blowups to happen, but it is still an open problem if such blowups occur in free space R^3 . Special cases of this problem have been solved before; in particular, Hou and Chen established in 2022 the presence of a blowup in the Euler system when there is boundary instead of free space. The boundary helps confine the singularity and makes the analysis tractable. We now provide evidence that the Euler equations admit a self-similar finite-time blowup on R^3 without the need for any boundary or forcing. Our starting point is a physics-informed neural network PINN to come up with an approximate answer, and then to argue stability around that to complete the proof. The challenge so far has been that PINNs have not been successful in discovering singularities on the current problem. A common failure mode is PINNs converging to a trivial solution. We take special care to nudge our PINN to interesting regions through a combination of constraints, and we believe this is a novel way of making PINNs work for such hard optimization problems. After PINN delivers a solution, we refine it with spline representation to allow us to obtain tight certified bounds. This is because our goal is not just to merely stop at an approximate solution, but to bound it tightly for stability analysis. We also carry out a detailed study of the transport field of our approximate profile, and show it has promising properties of local outgoing flow, essential for proving linear damping, which is an important ingredient of overall stability. Broader Implications While much of the recent progress in AI-assisted mathematics has focused on the role of large language models LLMs in tasks such as theorem proving and conjecture testing, the central computational ingredient in this work is physics-informed optimization. The governing PDEs are incorporated directly into the optimization problem, allowing candidate singularity structures to be evaluated. We believe that such physics-informed and physics-centric AI are critical ingredients across many areas of research involving physical systems, and LLMs lack such physical grounding. More broadly, our group’s work has involved physics-informed learning and neural operators with widespread impact such as training the first AI-based high resolution weather model, simulating plasma in nuclear fusion, designing better medical devices, accelerating quantum chemistry and it is essential to create AI models that understand physics. PS: We were just informed by Terence Tao of an announcement https://mastodon.social/@tristanbuckmaster/117233413705701198 by Alpöge and Tristan Buckmaster on Euler equations with forcing. We have been working on the related topic of incompressible Euler equations on R^3 but with no forcing, and a different route to proof using PINNs. It is wonderful to see different perspectives on this problem.