Caltech team (Anandkumar) reports a stable self-similar singularity candidate for the unforced 3D Euler equations on R³, found with PINNs and LLM help
Awaiting review
The takeaway
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.
Status
- Claim
Awaiting review
- Our reporting
- Medium confidence
- Verification
- Unverified claim
- Importance
- 3 of 5
- Last verified
- 29 September 2026
Your AI and this story
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None of these four assistants can know about it. The closest, Claude Opus 5.5, stops 69 days before it.
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
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.
What is disputed or not yet verified
| Verification | Interval-arithmetic certification of the profile; partial Lean formalisation; stability conditional |
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Sources
4 sources from 3 sites. Numbers match the chips in the text.
4 sources: 3 primary, 1 reaction
Primary
- arXiv 2609.10867: Self-Similar Singularity of the Euler Equations on R³arxiv.org, paper
- arXiv 2609.10860: Stability Framework for the Singularity of the Euler Equations on R³arxiv.org, paper
- Anandkumar group page on the Euler resulttensorlab.cms.caltech.edu, official
Reactions
- Anima Anandkumar (guest post on Tao’s blog): Stable singularity of the Euler equations on R³terrytao.wordpress.com, discussion
Changes
- Filed. Marked pending because the arXiv abstracts describe the stability proof as conditional