An ML approach to resolution of singularities

Published: 18 Jun 2023, Last Modified: 17 Jul 2023TAGML2023 PosterEveryoneRevisions
Keywords: Machine learning, Reinforcement learning, Resolution of singularities, Symbolic Computations, Commutative Algebra
Abstract: The solution set of a system of polynomial equations typically contains ill-behaved, singular points. Resolution is a fundamental process in geometry in which we replace singular points with smooth points, while keeping the rest of the solution set unchanged. Resolutions are not unique: the usual way to describe them involves repeatedly performing a fundamental operation known as "blowing-up", and the complexity of the resolution highly depends on certain choices. The process can be translated into various versions of a 2-player game, the so-called Hironaka game, and a winning strategy for the first player provides a solution to the resolution problem. In this paper we introduce a new approach to the Hironaka game that uses reinforcement learning agents to find optimal resolutions of singularities. In certain domains, the trained model outperforms state-of-the-art selection heuristics in total number of polynomial additions performed, which provides a proof-of-concept that recent developments in machine learning have the potential to improve performance of algorithms in symbolic computation.
Type Of Submission: Proceedings Track (8 pages)
Submission Number: 7