Hilbert series, machine learning, and applications to physics Article Swipe

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Series (stratigraphy) Physics Fano plane Embedding Intersection (aeronautics) Dimension (graph theory) Artificial neural network Hilbert space Binary number Artificial intelligence Hilbert–Poincaré series Particle physics Pattern recognition (psychology) Algorithm Pure mathematics Machine learning Quantum mechanics Computer science Mathematics Arithmetic Biology Paleontology Engineering Aerospace engineering

Jiakang Bao , Yang‐Hui He , Edward Hirst , Johannes Hofscheier , Alexander Kasprzyk , Suvajit Majumder ·

YOU? · · 2022 · Open Access · · DOI: https://doi.org/10.1016/j.physletb.2022.136966 · OA: W3139110513

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 three-dimensional Fano varieties considered.