Revolutionizing Mathematical Discovery: Uncovering New Insights with Large Language Models

Introduction

Large Language Models (LLMs) have revolutionized the field of natural language processing, enabling computers to understand and generate human-like text. However, their potential goes beyond language processing. In this article, we explore the use of LLMs to discover new mathematical knowledge and algorithms in different domains.

Driving discovery through evolution with language models

FunSearch, a method developed by our team, uses an evolutionary approach to harness the creativity of LLMs. By pairing a pre-trained LLM with an automated evaluator, we can identify and build upon the LLM’s best ideas, leading to new discoveries in mathematics and computer science.

Our team has successfully applied FunSearch to two challenging problems: the cap set problem, a longstanding open problem in mathematics, and the online bin-packing problem, a practical challenge in computer science. In this article, we will explore the results of these applications and discuss the potential of FunSearch for future scientific discoveries.

Breaking new ground in mathematics

We first addressed the cap set problem, an open challenge that has vexed mathematicians for decades. By using FunSearch, we discovered new solutions that were far more conceptually rich than previous attempts. The solutions generated by FunSearch were not only novel but also provided actionable insights for researchers.

What’s more, the interpretability of FunSearch’s programs can provide actionable insights to researchers. As we used FunSearch, we noticed intriguing symmetries in the code of some of its high-scoring outputs. This gave us a new insight into the problem, and we used this insight to refine the problem introduced to FunSearch, resulting in even better solutions.

Addressing a notoriously hard challenge in computing

Encouraged by our success with the theoretical cap set problem, we decided to explore the flexibility of FunSearch by applying it to an important practical challenge in computer science. The online bin-packing problem looks at how to pack items of different sizes into the smallest number of bins. It sits at the core of many real-world problems, from loading containers with items to allocating compute jobs in data centers to minimize costs.

The online bin-packing problem is typically addressed using algorithmic rules-of-thumb (heuristics) based on human experience. However, finding a set of rules for each specific situation – with differing sizes, timing, or capacity – can be challenging. Despite being very different from the cap set problem, setting up FunSearch for this problem was easy. FunSearch delivered an automatically tailored program that outperformed established heuristics, using fewer bins to pack the same number of items.

LLM-driven discovery for science and beyond

FunSearch demonstrates that if we safeguard against LLMs’ hallucinations, the power of these models can be harnessed not only to produce new mathematical discoveries but also to reveal potentially impactful solutions to important real-world problems.

We envision that for many problems in science and industry – longstanding or new – generating effective and tailored algorithms using LLM-driven approaches will become common practice.

Conclusion

FunSearch is a powerful tool for discovering new mathematical knowledge and algorithms in different domains. By harnessing the creativity of LLMs and using an evolutionary approach, we can identify and build upon the LLM’s best ideas, leading to new discoveries and insights. As we continue to develop and refine FunSearch, we are excited to explore its potential for future scientific discoveries and practical applications.