AI aids mathematicians in solving diverse problems
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The AI tools created by Google DeepMind are proving to be remarkably effective in aiding mathematical research, and experts believe this could initiate a wave of AI-driven mathematical breakthroughs on an unprecedented scale.
In May, Google unveiled an AI system named AlphaEvolve, which may reveal new algorithms and formulas. This system generates numerous potential solutions through Google’s AI chatbot Gemini, which then feeds them into a distinct AI evaluator. This evaluator filters out nonsensical outputs that chatbots are prone to produce. During initial tests, Google researchers pitted AlphaEvolve against over 50 unresolved mathematical problems, and discovered that it accurately rediscovered the most prominent solutions established by humans in approximately three-quarters of the cases.
Recently, Terrence Tao and his team at UCLA assessed the system using 67 more rigorous and extensive mathematical research queries. They found that AlphaEvolve did more than merely revisit old solutions; in certain instances, it could generate improved resolutions suitable for integration into other AI systems, like a more resource-intensive version of Gemini or AlphaProof, the AI that secured a gold medal in this year’s International Mathematics Olympiad, to craft new mathematical proofs.
Tao noted that it’s challenging to gauge overall effectiveness, as the problems differ in their complexities. However, the system consistently operated much faster than any individual mathematician.
“Addressing these 67 problems through traditional methods would require us to design a specific optimization algorithm for each task. That would take years and we might never have initiated this project at all. This initiative offers a chance to engage in mathematics on a previously unseen scale,” Tao states.
AlphaEvolve is particularly adept at solving what are known as optimization problems. These encompass tasks like determining the optimal figures, formulas, or objects that best resolve specific challenges. For instance, calculating the maximum number of hexagons that can occupy a defined area.
While the system is capable of addressing optimization problems across various branches of mathematics, such as number theory and geometry, these still represent “only a small fraction of all the problems that mathematicians are interested in,” according to Tao. Nonetheless, the power of AlphaEvolve is such that mathematicians might attempt to reformulate non-optimization problems into solvable forms for AI. “These tools offer a fresh perspective for tackling these issues,” he adds.
A potential drawback, however, as Tao explains, is that the system sometimes tends to “cheat” by producing answers that seem correct but utilize loopholes or methods that don’t genuinely solve the problems. “It’s akin to administering a test to a group of exceptionally bright yet morally ambiguous students who will do whatever it takes to score highly,” he remarks.
Even with its flaws, AlphaEvolve’s achievements are garnering interest from a broader segment of the mathematical community that might have previously leaned towards more general AI solutions such as ChatGPT, according to team member Javier Gomez Serrano from Brown University. Although AlphaEvolve isn’t publicly accessible yet, numerous mathematicians have expressed interest in testing it.
“There’s definitely a growing curiosity and openness to employing these tools,” asserts Gomez Serrano. “Everyone is eager to discover their potential. Interest in the mathematical community has surged compared to a year or two ago.”
Tao believes that such AI systems alleviate some of the burdens of mathematical work, allowing researchers to focus on other areas. “Mathematicians are few in number globally, making it infeasible to consider every problem. However, there exists a multitude of mid-level difficulties where tools like AlphaEvolve are particularly effective,” he notes.
Jeremy Avigado, a researcher at Carnegie Mellon University in Pennsylvania, observes that machine learning methods are increasingly beneficial to mathematicians. “The next step is enhancing collaboration between computer scientists skilled in machine learning tools and mathematicians with domain-specific knowledge,” he emphasizes.
“We aspire to witness more outcomes like this in the future and identify methods to extend this approach into more abstract mathematical fields.”
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Source: www.newscientist.com
