Hilbert Series, Machine Learning, and Applications to Physics

Abstract

We describe how simple machine learning methods successfully predict geometric properties from Hilbert series (HS). Regressors predict embedding weights in projective space to ∼1 mean absolute error, whilst classifiers predict dimension and Gorenstein index to > 90% accuracy with ∼0.5% standard error. Binary random forest classifiers managed to distinguish whether the underlying HS describes a complete intersection with high accuracies exceeding 95%. Neural networks (NNs) exhibited success identifying HS from a Gorenstein ring to the same order of accuracy, whilst generation of "fake" HS proved trivial for NNs to distinguish from those associated to the threedimensional Fano varieties considered.

Some open questions in quiver gauge theory

“…Its use initiated in this field with the examination of string landscapes [125][126][127][128], and has since quickly developed these techniques to a wide range of subfields related to gauge theories. In particular, ML has seen great success in topics discussed in this paper, examining: plethystics [129,130], amoebae [131], Seiberg duality [132], and dessins d'enfants [133].…”

Some Open Questions in Quiver Gauge Theory

“…Its use initiated in this field with the examination of string landscapes [122][123][124][125], and has since quickly developed these techniques to a wide range of subfields related to gauge theories. In particular, ML has seen great success in topics discussed in this paper, examining: plethystics [126,127], amoebae [128], Seiberg duality [129], and dessins d'enfants [130].…”

Brain Webs for Brane Webs

“…Neural Networks (NNs) are a primary tool within supervised ML, whose application on labelled data acts as a nonlinear function fitting to map inputs to outputs, both represented as tensors over Q using decimals. In recent years the advancement of computational power has played perfectly into the hands of these many-parameter techniques, leading to a programme of application of these tools to datasets arising in theoretical physics [18][19][20][21][22][23][24][25][26][27][28] and the relevant mathematics [29][30][31][32][33][34][35][36]. Motivated by this, we initiate the program of applying ML techniques to the classification of 5-brane webs and 5d SCFTs, concentrating on the simplest case of webs with exactly three external legs.…”