PatternBoost: Constructions in Mathematics with a Little Help from AI

TL;DR

PatternBoost is an innovative AI-driven method that combines classical search and transformer neural networks to discover new mathematical constructions, solving longstanding problems in extremal combinatorics.

Contribution

The paper introduces PatternBoost, a novel iterative approach integrating classical search with neural networks to find interesting mathematical constructions.

Findings

Found best known solutions to several long-standing problems.

Successfully constructed a counterexample to a 30-year-old conjecture.

Demonstrated impressive performance on various extremal combinatorics problems.

Abstract

We introduce PatternBoost, a flexible method for finding interesting constructions in mathematics. Our algorithm alternates between two phases. In the first ``local'' phase, a classical search algorithm is used to produce many desirable constructions. In the second ``global'' phase, a transformer neural network is trained on the best such constructions. Samples from the trained transformer are then used as seeds for the first phase, and the process is repeated. We give a detailed introduction to this technique, and discuss the results of its application to several problems in extremal combinatorics. The performance of PatternBoost varies across different problems, but there are many situations where its performance is quite impressive. Using our technique, we find the best known solutions to several long-standing problems, including the construction of a counterexample to a conjecture that had remained open for 30 years.