Will AI crack the hardest still unresolved mathematical problems to discover even harder, never before contemplated problems? 😊
I am having already a lot of fun learning about it!
From the abstract:
"Large Language Models (LLMs) have demonstrated tremendous capabilities in solving complex tasks, from quantitative reasoning to understanding natural language. However, LLMs sometimes suffer from confabulations (or hallucinations) which can result in them making plausible but incorrect statements. This hinders the use of current large models in scientific discovery. Here we introduce FunSearch (short for searching in the function space), an evolutionary procedure based on pairing a pre-trained LLM with a systematic evaluator. We demonstrate the effectiveness of this approach to surpass the best known results in important problems, pushing the boundary of existing LLM-based approaches. Applying FunSearch to a central problem in extremal combinatorics — the cap set problem — we discover new constructions of large cap sets going beyond the best known ones, both in finite dimensional and asymptotic cases. This represents the first discoveries made for established open problems using LLMs. We showcase the generality of FunSearch by applying it to an algorithmic problem, online bin packing, finding new heuristics that improve upon widely used baselines. In contrast to most computer search approaches, FunSearch searches for programs that describe how to solve a problem, rather than what the solution is. Beyond being an effective and scalable strategy, discovered programs tend to be more interpretable than raw solutions, enabling feedback loops between domain experts and FunSearch, and the deployment of such programs in real-world applications."
Mathematical discoveries from program search with large language models (early draft, no public access)
The FunSearch process. The LLM is shown a selection of the best programs it has generated so far (retrieved from the programs database), and asked to generate an even better one. The programs proposed by the LLM are automatically executed, and evaluated. The best programs are added to the database, for selection in subsequent cycles. The user can at any point retrieve the highest-scoring programs discovered so far.
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