OpenAI has announced that its internal AI system has made significant progress by addressing the existence and smoothness problem associated with the Navier–Stokes equations. This problem is one of the seven Millennium Prize Problems for which the Clay Institute has pledged a reward of $1 million for each solution. To date, only one of these problems has been resolved; the prize for the proof of the Poincaré conjecture was awarded to Grigori Perelman in 2010.
The Navier–Stokes equations are fundamental in fluid dynamics, representing Newton's second law in the context of continuous media, such as liquids and gases. Historically, a comprehensive solution has eluded mathematicians and engineers, who typically rely on numerical methods or specific case formulas. The central query remains: if a fluid begins its motion smoothly and possesses finite energy, will it maintain that smooth flow indefinitely, or will there come a moment when the flow becomes infinite? An infinite velocity would indicate a breakdown in the equations' applicability to real fluids.
OpenAI's model has presented an example where a flow, starting from rest and influenced by a smooth, bounded external force, accelerates to infinite speed in a finite time frame while maintaining finite energy. This phenomenon resembles a vortex spiraling inward, elongating like a strand of spaghetti, with its central area contracting and accelerating without bounds simultaneously.
Interestingly, OpenAI does not intend to pursue the $1 million reward. The Clay Institute has stringent guidelines that include publication in a peer-reviewed journal, a minimum two-year waiting period, and community validation of the solution. OpenAI emphasizes that this achievement is more a demonstration of AI's capabilities rather than a bid for the prize. Despite the formalization of the proof being published in the Lean language, it will undergo scrutiny from mathematicians globally.
The situation has become more complex due to comments from Tristan Buckmaster, a mathematician from New York University. Along with Levent Alpöge from Anthropic, he has issued three proofs related to the breakdown of solutions for similar equations. Buckmaster claimed that OpenAI suggested he release the Navier–Stokes findings under his name alone, excluding Alpöge due to his affiliation with a competing organization. The mathematicians had worked on their drafts within Codex sessions, and while Buckmaster inquired if OpenAI's model had been trained on their work, he received no clarification. He maintains that he has not reviewed OpenAI's proof and is not making accusations. Sébastien Bubeck, mentioned in Buckmaster's statement, has already rebutted these claims as false and has promised a comprehensive response.
Is this a significant advancement? Many are celebrating it as a landmark day for AI, while others argue that the AI may have appropriated the mathematicians' contributions.
Informational material. 18+.
