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OpenAI AI: Navier-Stokes Smoothness Breakdown Demonstrated

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OpenAI AI: Navier-Stokes Smoothness Breakdown Demonstrated

Clip title: The Equation That Might Destroy Itself Author / channel: Two Minute Papers URL: https://www.youtube.com/watch?v=mOvtumfyjCs

Summary

The video discusses the groundbreaking announcement by OpenAI regarding a likely solution to the Navier-Stokes existence and smoothness problem, one of the seven Millennium Prize Problems. Károly Zsolnai-Fehér, the host and a research scientist in fluid simulations, expresses his disbelief and excitement over this potential historical breakthrough. He notes that while there is some controversy, primarily around the proprietary nature of OpenAI’s systems and potential data reuse from other scientists, the achievement itself is monumental.

To provide context, Zsolnai-Fehér breaks down the Navier-Stokes equations, which describe fluid motion. He explains the three core terms: advection (how fluid carries itself), pressure (forces from fluid density), and diffusion (how differences average out). Coupled with the incompressibility condition (constant volume), these equations model complex fluid behaviors. The central question of the Millennium Prize problem is whether, for a smooth initial fluid flow, the mathematics of these equations will always remain smooth or if they can “break down” (develop singularities, like infinitely high velocity) within a finite time, even if the total energy remains finite.

OpenAI’s internal AI model has reportedly found a specific scenario that answers this question. By carefully creating a vortex that spirals inward, stretches, and increases in velocity without bound in a finite amount of time, while still maintaining finite total energy, the AI system demonstrated that the mathematical “smoothness” can indeed break down. This result indicates that solutions to the Navier-Stokes equations are not always smooth, providing a negative answer to the existence and smoothness question as it pertains to singularities.

A remarkable aspect highlighted in the video is the speed of this discovery. OpenAI’s agents reportedly reached this resolution in approximately 88 hours after being launched. Zsolnai-Fehér attributes AI’s rapidly accelerating prowess in mathematics to its verifiability. Unlike creative tasks, mathematical proofs can be checked automatically at an incredibly high rate (hundreds of millions of times per hour by a robot versus hundreds by a human), allowing AI to rapidly learn and validate complex mathematical concepts. This immense computational advantage enables AI to explore vast solution spaces and uncover proofs far more quickly than human mathematicians.

The video concludes with a strong sense of optimism for AI’s future impact on scientific discovery. Zsolnai-Fehér shares an anecdote about DeepMind’s Demis Hassabis envisioning cures for all diseases within a decade by treating them as verifiable problems, akin to mathematics. This highlights the transformative potential of AI to solve previously intractable problems across various scientific domains, making it “what a time to be alive” for scientific and technological advancements.

Video Description & Links

Description

📝 The Navier-Stokes solution paper is available here: https://openai.com/index/navier-stokes-solution/

My fluid simulations and papers: https://users.cg.tuwien.ac.at/zsolnai/gfx/fluid_control_msc_thesis/ https://users.cg.tuwien.ac.at/zsolnai/gfx/real_time_fluid_control_eg/ The flow from simulation to reality: https://www.nature.com/articles/s41567-022-01788-5 All papers: https://users.cg.tuwien.ac.at/zsolnai/

Full interview with Sir Demis Hassabis: https://www.youtube.com/watch?v=huAwz_BR8WM

Sources: https://www.youtube.com/watch?v=EURkO98VnKc https://www.youtube.com/watch?v=luOWq1Gdv8c

Adam Bridges, B Shang, Carlos Galarza, Christian Ahlin, Eric Tyson, Juan Benet, Lukas Biewald, Michael Tedder, Owen Skarpness, Ryan Stankye, Shawn Becker, Steef, Taras Bobrovytsky, Tazaur Sagenclaw, Tybie Fitzhugh, Ueli Gallizzi

Tags

ai, navier-stokes, openai, millennium problem, navier stokes ai

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