Caltech team (Anandkumar) reports a stable self-similar singularity candidate for the unforced 3D Euler equations on R³, found with PINNs and LLM help
On 7 Sep 2026, the evening before OpenAI's Navier–Stokes announcement, Anima Anandkumar's Caltech group posted a self-similar singular profile for the unforced incompressible 3D Euler equations on all of R³. Physics-informed neural networks found it, and LLMs helped simplify the bounds and formalise derivations in Lean. The arXiv papers (2609.10867, 2609.10860) describe "evidence" and a stability framework that is conditional on certifying explicit constants, so this is not yet a complete proof.
Key facts
- Authors: Adarsh Ganeshram, Valentin Duruisseaux, Anima Anandkumar (+ Robert J. George on the stability paper)
- Setting: incompressible Euler on unbounded R³, no forcing; axisymmetric self-similar ansatz at blow-up rate 0.5 (matching a prediction by Constantin et al., arXiv 2602.17570)
- Method: PINN finds approximate profile; second-order optimisers (SS-eSOAP, SS-Broyden); certified via spline representation with interval arithmetic
- AI use (guest post): 'we used the OpenAI and other models extensively to simplify our bounds as well as formalize the derivations in Lean'
- arXiv 2609.10867 (111 pp.) abstract: 'We provide evidence of a finite-time singularity'; 2609.10860 (113 pp.): stability proof closes 'conditional on rigorous certification of the estimates and constants'
- The authors complain that mainstream media followed OpenAI's press release and did not acknowledge their work
Science result
- Field
- mathematics / partial differential equations / fluid dynamics
- Problem
- Finite-time singularity for the unforced 3D incompressible Euler equations on R³ from smooth initial data
- Result
- Numerically certified self-similar singular profile plus a (conditional) framework for its nonlinear stability; full rigorous blow-up proof not yet complete.
- AI system
- physics-informed neural networks, OpenAI models and other LLMs
- Human role
- Human-led; AI (PINNs) discovered the candidate, LLMs simplified bounds and helped formalise derivations
- Verification
- Interval-arithmetic certification of the profile; partial Lean formalisation; stability conditional
- Status
- pending
- Why surprising
- Unlike OpenAI's and Buckmaster–Alpöge's results, it targets the unforced problem on the whole space, which is closer to what physicists care about.
What happened
In the same week as the Buckmaster–Alpöge forced blow-up results (7 Sep) and OpenAI's forced Navier–Stokes claim (8 Sep), a third group posted a singularity for the unforced Euler equations on the whole space. They used AI-driven numerical discovery followed by computer-assisted proof techniques. Their guest post on Tao's blog stresses AI as a "complementary" tool, "built to propose solutions that did not compete with humans".
Why it matters
Unforced Euler blow-up on R³ is a famous open problem in its own right, and it is a stepping stone toward the unforced Navier–Stokes question. The claim is still partly conditional, so its status should be tracked.
Changelog
- 2026-09-29: created (lead from data/leads.md); marked pending because the arXiv abstracts describe the stability proof as conditional
Related events
- OpenAI claims a Millennium Prize problem: 10,000 AI agents prove forced Navier–Stokes blow-up; priority dispute erupts ★★★★★
- DeepMind and mathematicians use neural networks to find new unstable singularities in fluid equations ★★★
Sources (4)
- discussionAnima Anandkumar (guest post on Tao's blog): Stable singularity of the Euler equations on R³
- paperarXiv 2609.10867: Self-Similar Singularity of the Euler Equations on R³
- paperarXiv 2609.10860: Stability Framework for the Singularity of the Euler Equations on R³
- officialAnandkumar group page on the Euler result
id: 2026-09-07-anandkumar-euler-singularity-r3 · updated 2026-09-29 · open in the interactive timeline