# Accelerated Understanding’s Team Finds Stable Singularity In Euler Equations, Terrance Tao Reacts > Source: > Published: 2026-09-08 13:29:04+00:00 The AI revolution had started off with advances in coding, but the action now seems to moved to math. Caltech professor Anima Anandkumar has announced a new result on one of fluid dynamics’ oldest open questions, and the timing could not have been stranger: it landed on the very same day as a separate, unrelated [breakthrough](https://officechai.com/ai/mathematician-tristan-buckmaster-says-he-cracked-a-fluid-dynamics-problem-with-ai-accuses-openai-of-trying-to-take-credit/) on a closely related problem from a Anthropic-affiliated mathematician, prompting Fields Medalist Terence Tao to publicly note the coincidence. In a post on her blog, Anandkumar, working with collaborators Ganeshram and Duruisseaux, laid out evidence for a “stable singularity” in the 3D Euler equations, the simpler, viscosity-free cousin of the Navier-Stokes equations that describe how fluids like air and water move. Both equations belong to a small club of problems that have resisted a complete mathematical description for centuries, and the question of whether they can spontaneously “blow up” — develop a point where a physical quantity spirals to infinity in finite time, even from smooth, well-behaved starting conditions — sits right at the heart of one of the seven Millennium Prize Problems. ## What the team actually did The approach is notable less for the specific equation it tackles and more for how the search was carried out. Rather than using a large language model to hunt for a proof, Anandkumar’s team built what’s called a physics-informed neural network, or [PINN](https://officechai.com/ai/accelerated-understanding-comes-out-of-stealth-to-simulate-and-discover-physics/) — a type of AI model that is trained not on text or images, but on the governing equations of a physical system, so its output is constrained to obey the laws of physics rather than just fit data. Getting a PINN to actually find something interesting turned out to be its own obstacle. According to the post, PINNs applied to this problem have historically had a habit of settling on the “trivial solution” — the equivalent of an optimizer shrugging and returning a boring, uninteresting answer rather than a genuine singularity. Anandkumar’s team says it added a specific combination of constraints to nudge the network away from that lazy outcome and toward the more exotic regions of the solution space where a real blow-up candidate might live. Once the PINN produced a candidate singularity shape, the team refined it using spline representations to get tight, certifiable bounds around it, since an approximate answer alone isn’t a proof — a rigorous argument needs to show the shape is stable, meaning small errors around it don’t spiral out of control. The blog post says the team’s analysis of the “transport field” around their candidate profile shows promising signs of the kind of local outgoing flow needed to demonstrate that stability, though the full proof of stability is still described as a work in progress. Anandkumar used the occasion to make a broader argument about AI’s role in this kind of research: that physics-centric, physics-informed AI systems are doing indispensable work in areas involving physical systems, and that general-purpose LLMs, however capable at proof-writing, don’t have that kind of physical grounding built in. ## An “independent” collision with a rival result The post carries an unusual postscript. Just as it was going up, Anandkumar’s team learned from Tao himself that mathematician Tristan Buckmaster and Anthropic’s Levent Alpöge had released their own result on a version of the Euler equations — one that uses forcing, an added external term, rather than the unforced, free-space version Anandkumar’s group targeted. That announcement, which we covered [when Buckmaster and Alpöge first published their Lean-verified proofs](https://officechai.com/ai/terrance-tao-calls-buckmaster-alpoges-fluid-dynamics-proofs-a-remarkable-achievement-says-could-help-solve-navier-stokes/), had already drawn Tao’s praise as a “remarkable achievement,” and separately spiraled into a public dispute between [Anthropic and OpenAI researchers over credit](https://officechai.com/ai/anthropic-openai-researchers-spar-over-what-appears-to-be-credit-for-progress-towards-solving-navier-stokes-equation/) for progress toward Navier-Stokes. Tao, weighing in on Anandkumar’s result separately, called it “by sheer coincidence” that two independent results on the Euler blow-up question surfaced at once, describing Anandkumar’s approach as the more “mainstream” route to finite-time blow-up: using numerical simulation or machine learning to locate a plausible blow-up template, or ansatz, rather than reasoning it out symbolically. He noted the candidate profile the team found appears numerically stable, but cautioned that turning that into an actual rigorous proof of stability is still missing and could demand a large amount of further computational and theoretical work. Some of the required properties have, however, already been partially formalized in the Lean proof assistant. Tao also pointed out that this result leans far less on AI for its core reasoning than some other recent papers in the space, with AI’s role here largely confined to literature review and Lean formalization rather than PINN-driven design — a contrast to Anandkumar’s own PINN, which sits at the very center of her approach. The dual announcements arrive amid a broader reckoning over what AI-assisted proofs of landmark problems actually mean for mathematics as a field. Tao [recently argued](https://officechai.com/ai/terrance-tao-explains-how-ai-powered-math-proofs-could-be-a-net-negative-for-math/) that if such a “solve” arrives as a sealed black box, with the method kept hidden, it could end up poisoning a problem as a source of future progress rather than energizing it — a concern that puts extra weight on how openly groups like Anandkumar’s and Buckmaster’s document their methods going forward. Anandkumar’s use of physics-informed models for this result also lines up with the pitch behind her newest venture. She recently [co-founded Accelerated Understanding](https://officechai.com/ai/accelerated-understanding-comes-out-of-stealth-to-simulate-and-discover-physics/) with AI infrastructure engineer Benedikt Jenik, a startup built around the idea that predicting how physical systems evolve in space and time, rather than predicting the next word in a sentence, is the more useful frontier for AI to push on next.