‘Didn’t even plan to publish it’: Indian scientist on 2017 paper cited in OpenAI’s proof of $1 million math problem

However, Ethan Vishniac, Editor-in-Chief at American Astronomical Society, convinced Singh to publish the result over dinner in Bangalore in 2010. Singh says, “Vishniac told me that this is an exact solution, you must definitely publish it. There are not many exact solutions.”

The first attempt at publication failed after one of the four referees rejected the paper. Finally in 2017 the result was published on invitation in The European Physical Journal Plus with the title Plane shearing waves of arbitrary form: Exact solutions of the Navier-Stokes equations.

What is the Navier-Stokes problem?

The Navier-Stokes equations are a set of partial differential equations concerned with the flow of fluids. They are used in aircraft design, weather forecasting, modelling blood circulation, pipelines, and all such areas of natural flow.

Within these larger equations, a specific problem called the ‘existence and smoothness’ problem is one of the seven Millennium Prize Problems announced by the Clay Mathematics Institute of Cambridge in 2000. These seven problems carry a $1 million prize.

If OpenAI’s claim is confirmed to be true, it will be the second Millennium Prize Problem to be solved after Grigoriy Perelman solved the Poincaré Conjecture in 2003. Mathematicians are rushing to verify OpenAI’s results.

On his own work specifically, Singh said that he had been interested in astrophysical flows also governed by similar Navier-Stokes equations and had published one such exact smooth flow solution in his 2017 paper. OpenAI, on the other hand, has done much more ‘comprehensive analysis’, he says.

Reaction to OpenAI’s solution

Like many other scientists, Singh is not a fan of the way in which the problem has been solved. He remarks, “It is depressing that a machine has managed to solve this whereas so many scientists for centuries have been trying to do this. When a human makes an attempt and it fails, each of those attempts teaches something unique. But if suddenly somebody gives you a dry solution, I don’t know how useful it becomes.”

But what about the practical scientific applications that solutions provided by AI might result in? Singh opines, “I am not necessarily an optimist. But an optimist would take a different point of view and I can understand and sympathise with that point of view also. He would say ‘Let’s go full force and apply this and come up with better weather prediction tools.’ One can always argue like a marketing agent.”

Terrance Tao, professor at UCLA and one of the most celebrated mathematicians in the world, similarly posted on Mastodon, “the indiscriminate use of powerful solution-extraction tools can achieve the immediate short-term goal of solving problems at hand, but at the cost of sustaining the ecosystem for the next wave of progress, or in understanding the progress already obtained.”

OpenAI’s announcement has also come with significant controversy for another reason. The AI company has accepted that it started working on the problem after hearing rumours that rival AI company Anthropic had solved the problem.

Mathematicians Levent Alpöge, an Anthropic employee, and Tristan Buckmaster said that they had also been working on a part of the problem using OpenAI’s codex, meaning the company might have used their data to train its model. OpenAI does not deny this and said in a statement, “While unlikely, we cannot rule out that de-identified data derived from their usage of our products helped improve our models.”