Sergiu Klainerman is an expert in PDEs and highly qualified to speak on both the Navier-Stokes problem and the future of mathematics. I was privileged to have him as my PhD advisor.
Despite the title, a good part of his essay addresses the formulation of the Navier-Stokes problem and whether it might have been a bit "too easy". If I could summarize in my own words, there are roughly three problems of increasing difficulty.
1. Forced: you're allowed to "stir the fluid" to try to produce an infinite vortex. This is what OpenAI accomplished, building on the works of others.
2. Unforced: no stirring allowed - can you construct a calmer fluid state at time t=0 so that an infinite vortex forms at a later time t=T? (Or equivalently at least stir the fluid and then let go before the infinite vortex forms.) Alternatively, prove that to be impossible.
3. General: understand and categorize conditions that lead to an infinite vortex, showing (most likely) that such a phenomenon arises only from contrived examples.
AI is pretty good at counterexamples. It will be interesting to see if AI can make progress on the general problem, which likely requires understanding of Navier-Stokes at a fundamentally deeper level.
Founderarcstone•56m ago
Maybe math evolves and humans are elevated from Ai? Hard to say!
krupan•49m ago
Unfortunate title when this is actually a better explanation of what OpenAI actually did and did not with Navier-Stokes
jstogin•1h ago
Despite the title, a good part of his essay addresses the formulation of the Navier-Stokes problem and whether it might have been a bit "too easy". If I could summarize in my own words, there are roughly three problems of increasing difficulty.
1. Forced: you're allowed to "stir the fluid" to try to produce an infinite vortex. This is what OpenAI accomplished, building on the works of others.
2. Unforced: no stirring allowed - can you construct a calmer fluid state at time t=0 so that an infinite vortex forms at a later time t=T? (Or equivalently at least stir the fluid and then let go before the infinite vortex forms.) Alternatively, prove that to be impossible.
3. General: understand and categorize conditions that lead to an infinite vortex, showing (most likely) that such a phenomenon arises only from contrived examples.
AI is pretty good at counterexamples. It will be interesting to see if AI can make progress on the general problem, which likely requires understanding of Navier-Stokes at a fundamentally deeper level.