TL;DR

UCLA Professor Ernest Ryu solved a 40-year-old optimisation theory problem—the stability mystery of the Nesterov Accelerated Gradient method—using GPT-5 as a collaborative research tool. What might have taken weeks of focused work was accomplished in approximately 12 hours over three days.

The Problem That Wouldn’t Yield

Since 1983, mathematicians have wondered why the Nesterov Accelerated Gradient (NAG) method delivers dramatically faster algorithm performance without introducing instability. Unlike a car engine that might fail when pushed too hard, NAG provides a speed boost while remaining remarkably stable—but nobody could prove why.

Ryu, with 15 years in applied mathematics and optimisation theory, had attempted this problem before without success. After testing ChatGPT-3.5 in 2023 and finding it wanting, he revisited the challenge when GPT-5 emerged with enhanced mathematical capabilities.

A New Kind of Research Partnership

What made GPT-5 different wasn’t its ability to solve the problem independently—it couldn’t. Instead, Ryu describes the model as an unusually creative collaborator that could rapidly surface techniques from across mathematical literature.

“GPT-5 was a very unusual collaborator in that it would propose something completely out of the blue,” Ryu explained. The model excelled at what humans find cognitively exhausting: rapidly proposing and discarding variations of ideas whilst borrowing tools from adjacent mathematical subfields.

The dynamic resembled exploring a massive maze with a companion who could reveal new paths instantly. Most paths led nowhere, but the speed of exploration meant dead ends were discovered quickly. The turning point came when GPT-5 suggested restructuring the governing equations—a suggestion that was incorrect as written but contained a structural insight Ryu developed into the final proof.

Verification Remains Human

Despite GPT-5’s contributions, the process demanded constant human oversight. Ryu found greater success starting fresh conversations rather than asking the model to check its own work, and he meticulously verified every step. “Several of the key steps that ultimately mattered were suggested by GPT-5,” he noted, “even though it could not assemble them into a complete proof on its own.”

Looking Forward

Ryu’s pre-print paper is now in peer review. He’s credited GPT-5 as a tool rather than co-author, conforming to traditional academic standards whilst transparently acknowledging its contributions throughout.

“This experience really had a profound influence on me. I plan to use AI in my math research all the time going forward,” Ryu concluded. “There’s no reason why I wouldn’t.”

Source: OpenAI