OpenAI has announced a potential solution to the Navier-Stokes equations, a complex Millennium Prize Problem, using advanced AI models.
OpenAI declared on September 8 that it has made significant strides in solving the Navier-Stokes equations, a notorious challenge recognized by the Clay Mathematics Institute as one of the Millennium Prize Problems, which carries a $1 million award.
We’re sharing a solution to the Navier-Stokes Millennium Prize Problem, one of the deepest problems at the frontier of mathematics.
The proof was produced by a group of agents using an OpenAI model that surpasses GPT-6 Astra in capabilities.
The Navier-Stokes equations are essential in understanding fluid dynamics, specifically regarding the persistence of smooth three-dimensional flow over time. OpenAI’s approach involved creating an example where the flow loses its smoothness due to external forces.
“The solution represents a vortex—a swirling fluid flow that tightens and elongates like spaghetti,” OpenAI explained.
This finding indicates a compression in the central region of the fluid as its velocity increases indefinitely, which helps address certain aspects of the problem's official criteria.
Following an internal model training on August 28, OpenAI's team decided to investigate the Millennium Problems after observing potential breakthroughs from other researchers in hydrodynamics. The system operated with approximately 10,000 agents who communicated extensively—over 2.7 million messages while generating around 130 billion output tokens during the Navier-Stokes analysis.
The agents completed their work on September 5, taking about 88 hours for the primary solution and an additional 17 hours to formalize and verify it with GPT-6 Astra.
For a solution to be considered valid by the Clay Mathematics Institute, it needs peer-reviewed publication and acknowledgment by the mathematical community for a minimum of two years. To date, only the Poincaré conjecture has been officially solved among the Millennium Problems.
OpenAI has stated it will not pursue the monetary prize, focusing instead on showcasing the capabilities of its AI models:
“Our goal in publishing this result is to report significant progress made by our AI models. We do not intend to claim the prize for this result.”
Contest for Discovery Priority
This announcement comes at a time when researchers such as Professor Tristan Buckmaster from New York University and Levent Alpöge from Harvard have reported their results on singularity formation under smooth external forces relevant to the same equations. They had shared their findings just a day before OpenAI’s announcement.
OpenAI acknowledged an initial misunderstanding regarding the nature of their solution but clarified that they did not access the unpublished work of Buckmaster and Alpöge. Nonetheless, OpenAI noted the theoretical possibility that anonymized data from their interactions could have influenced their model's development.
We congratulate Levent Alpöge and Tristan Buckmaster on their remarkable mathematical work.
OpenAI emphasized that their solution is fundamentally different from the findings by these researchers.
ChatGPT Images 2.5 Launch
On the same day, OpenAI introduced ChatGPT Images 2.5, enhancing its image generation model with faster processing and improved accuracy for editing images.
ChatGPT Images 2.5—faster, sharper, smarter, with better tools for creating whatever you can dream of.
Among its new features is a Sketch tool that allows users to create sketches directly in the interface, improving the editing and generation process. The update promises to enhance fidelity and maintain detail better across multiple iterations.
