PINNs Surface a Candidate Singularity in 3D Euler Equations

Episode 1 · Anandkumar Team Proves Stable Singularity in 3D Euler Equations with PINN Assist (2026-09-08, 14 posts)

On 09-08, Caltech professor and former NVIDIA research director Anima Anandkumar's team announced the discovery and proof of a stable singularity in the 3D Euler equations, a long-standing open problem in the mathematical foundations of fluid dynamics. The approach was two-fold: first obtain an approximate solution near the singularity using physics-informed neural networks (PINNs), then prove local stability around that solution. The next day the team released a companion paper detailing the nonlinear stability analysis needed to complete the proof, updated their blog, and shared a Lean formalization; Anandkumar also publicly addressed controversies around the team's Navier-Stokes work, crediting team members including Adarsh Ganeshram.

Confirmed

• Anandkumar personally announced the result on 09-08 and the companion stability paper on 09-09, constituting primary sources
• The singularity is a stable self-similar finite-time singularity in free space R^3 with no boundaries or external forcing; DrSingularity noted certified numerical bounds support the stability argument
• PINNs previously failed on such problems by converging to trivial solutions; the team avoided this via constraint combinations
• The candidate requires no external forcing or boundary conditions, meaning the fluid itself can evolve increasingly extreme behavior until the smooth description breaks down
• Terry Tao publicly acknowledged the team's research (per Anandkumar's account)
• The proof has been formalized in Lean, and team contribution attribution was clarified amid the Navier-Stokes controversy

Unconfirmed

• DrSingularity framed the result as solving a major open problem "if the proof holds"; completeness and peer review remain pending, and claims of overnight independent verification lack specifics

Why it matters

• The 3D Euler singularity problem is central to fluid dynamics foundations and closely tied to the unsolved Navier-Stokes Millennium problem
• The result demonstrates the viability of a hybrid "AI finds approximate solution + rigorous math proves it" paradigm, addressing long-standing doubts about PINNs
• The public attribution dispute highlights that crediting contributions has become a community flashpoint for major AI-math announcements

• Anandkumar team proves stable singularity for 3D Euler using PINN-guided math proof — AnimaAnandkumar · 2026-09-08
• Anandkumar Team Uses PINNs to Find Stable Singularity in 3D Euler — basedjensen · 2026-09-08
• Anandkumar team proves stable singularity in 3D Euler using PINN-driven proof — AnimaAnandkumar · 2026-09-08
• Anima Anandkumar's team proves stable 3D Euler singularity using PINNs — cephaloform · 2026-09-08
• Anima Anandkumar's team proves stable singularity in 3D Euler using PINNs — Thom_Wolf · 2026-09-08
• Anima Anandkumar's team uses PINNs to find stable singularity candidate in 3D Euler equations — calabi_and_yau · 2026-09-08
• Anandkumar team finds stable singularity in 3D Euler using PINNs, independently verified — thesaraharminta · 2026-09-08
• AI may have found a real singularity in the 3D Euler equations — Dr_Singularity · 2026-09-09
• Researchers present evidence of a stable finite-time singularity in the 3D Euler equations on R^3 — Dr_Singularity · 2026-09-09
• Anima Anandkumar's Team Releases Companion Paper on PINN Nonlinear Stability Analysis — AnimaAnandkumar · 2026-09-09
• Stable singularity found on 3D Euler using PINNs, Anandkumar team reports — anshulkundaje · 2026-09-09
• AI may have found a real singularity in the 3D Euler equations — Dr_Singularity · 2026-09-09
• PINN discovers evidence of stable finite-time singularity in 3D Euler equations — Dr_Singularity · 2026-09-09
• Anima Anandkumar's Caltech team completes Navier-Stokes proof with Lean formalization — AnimaAnandkumar · 2026-09-09

Episode 2 · Anandkumar Uses PINNs to Find Euler Singularity Candidates (2026-09-11, 2 posts)

Caltech professor Anima Anandkumar published a guest post on Terence Tao's blog describing how her team used physics-informed neural networks (PINNs) to discover singularity candidates for the 3D Euler equations, showcasing AI's role in pure mathematics research.

• Anima Anandkumar guest posts on Terence Tao's blog: PINNs find Euler singularity candidates — AnimaAnandkumar · 2026-09-11
• Anima Anandkumar guest-posts on Terence Tao's blog: PINNs designed to discover singular candidates — AnimaAnandkumar · 2026-09-13