Navier-Stokes and the Future AI Acceleration in Science

As the news breaks¹² of a partial solution to the famous Navier-Stokes equations, one of the major heretofore unsolved problems in mathematics³, it’s worth taking stock of where (and how) this result lands for the promise of AI acceleration in science. While an exciting major milestone in mathematics and a triumph of frontier model capabilities⁴, I think this work marks just the beginning of the beginning of the acceleration of AI more broadly in science. Let me explain.

The Navier-Stokes equations describe how a fluid’s velocity evolves under its own inertia, pressure, and viscosity, and the Clay Millennium Prize asks whether smooth 3D solutions always exist for all time or can break down in finite time³. The problem statement allows either a proof of global regularity (options A/B) or a demonstration of finite-time blow-up with smooth external forcing (options C/D)³.

While a pure mathematical construct, NS in its most generic form captures many of the properties of real physical systems, based on simple real-world “axioms” like conservation of matter and conservation of momentum. As such, a wide group of physical scientists look to NS as a mathematical description of our scenarios: aerodynamicists studying airflow over wings, us astronomers studying the flow of ejecta from a supernova into the interstellar medium, climate scientists modeling the atmosphere and oceans, and biomedical engineers modeling blood flow through arteries. The unknown answer until today was whether fluid flow, regardless of initial conditions, would always be smooth or would lead to a blow-up “singularity” (not in the AI sense but in the physical sense, with increasingly more energy concentrated in ever smaller scales), where energy from large scales could get transferred to smaller and smaller scales without getting dissipated by the viscosity (or pressure) of the fluid. Understanding NS would help us understand turbulence.

Now it seems that there are indeed conditions where singularities can arise — proven, with Lean-checked proofs, for the inviscid Euler equations and two simpler model equations⁵, and, OpenAI claims, for Navier-Stokes itself¹ — but those are contrived, driven by an external driving force that is particularly tuned to the initial conditions and properties of the fluid. This is the “C/D” scenario of NS, and it certainly portends the possibility that such conditions could exist without a driving force (and that viscosity does not always save the day)⁶. Indeed, that possibility is now partly realized one rung down the ladder: as a stepping stone to the forced NS result, OpenAI’s agents also claim a finite-time singularity for the unforced Euler equations — no external force at all, with viscosity set to zero — again with a writeup and a Lean formalization⁷.

So where does this land in the AI-accelerated science landscape? First, the “C/D” scenario of NS is not physically plausible. While it may one day be set up in a lab to confirm it, highly tuned driving forces do not generally exist in nature. Second, and perhaps more fundamentally, NS was never a complete description of physical systems. On small scales, fluids do not behave according to the NS equations; instead, finite particle sizes become important (e.g., the mean free path of air molecules at sea level is about 68 nm⁸, below which there is no continuum to describe), so mathematically existent blow-up singularities may happen in regimes where the equations aren’t valid.

The frontier LLMs are clearly getting very good at the “Thinking” fields, where axioms can be posited, reasoned over, proven, and used to construct ever more complex edifices on 100% solid ground (thanks to rigorous languages like Lean 4). Examples of this working well have abounded over the past few months, even before today’s news: Claude agents produced the first end-to-end machine-checked Lean proof of Fermat’s Last Theorem in eleven days, a task the human community effort had budgeted years for⁹¹⁰; frontier models have produced a counterexample to a decades-old conjecture¹¹, a new derivation in theoretical physics¹², and a disproof of a statistics conjecture in about 90 minutes¹³; and between October 2025 and August 2026, 65 open Erdős problems received their first accepted full solution with AI involvement, 32 of them with no significant human math input¹⁴.

But axiomatic thinking is only a component of science. And for many sciences the major challenges arise not in proving but in observing and comparing. Figure 1 is meant to capture this progression. Theory-driven fields like quantum chemistry require physical axioms as well as coding and (massive) compute. Data-driven fields like astronomy require physical axioms, coding and compute, and confrontation with real data about the physical world. Curiosity-driven fields require all of that AND the acquisition of new data. The emerging frontier of science, really, is those places where no existing data in the world can help answer a fundamental question, and where we probably need new instruments to answer it. A good example here is the question of life on planets outside of our solar system, or the nature of dark matter — we can’t think or compute our way to an answer, and likely the data does not already exist, nor can it be obtained with existing instruments. For those questions, and others like them, we’ll need to build new instruments. (And there may be some questions we never get to answer no matter what new experiments we build, like whether the multiverse hypothesis is correct or whether the proton decays.)

I don’t think AI acceleration will happen sequentially from the bottom up — to be sure, AI is already having an impact in all of these realms. But I hope this is helpful framing, especially given the milestone news.

References & Further Reading