Disentangling a deep learned volume formula

Abstract

We present a simple phenomenological formula which approximates the hyperbolic volume of a knot using only a single evaluation of its Jones polynomial at a root of unity. The average error is just 2.86% on the first 1.7 million knots, which represents a large improvement over previous formulas of this kind. To find the approximation formula, we use layer-wise relevance propagation to reverse engineer a black box neural network which achieves a similar average error for the same approximation task when trained on 10% of the total dataset. The particular roots of unity which appear in our analysis cannot be written as e2πi/(k+2) with integer k; therefore, the relevant Jones polynomial evaluations are not given by unknot-normalized expectation values of Wilson loop operators in conventional SU(2) Chern-Simons theory with level k. Instead, they correspond to an analytic continuation of such expectation values to fractional level. We briefly review the continuation procedure and comment on the presence of certain Lefschetz thimbles, to which our approximation formula is sensitive, in the analytically continued Chern-Simons integration cycle.

Augmenting astrophysical scaling relations with machine learning: application to reducing the Sunyaev-Zeldovich flux-mass scatter

“…We then implement symbolic regression to find an analytic form for a cluster mass proxy using the pre-selected parameters. Wadekar et al (2020); Graham et al (2013Graham et al ( , 2012; Shao et al (2021); Liu & Tegmark (2020); Wilstrup & Kasak (2021); Lemos et al (2022); Butter et al (2021); Gilpin (2021); ; Cranmer et al (2021b,a); Craven et al (2021); Werner et al (2021).…”

(K)not machine learning

“…In this article, which summarizes the results of [21][22][23], we explore new relations between knot invariants. In Section 2, we introduce the relevant knot invariants.…”

Machine Learning Kreuzer--Skarke Calabi--Yau Threefolds

“…In certain cases, neural network based curve fitting has facilitated the search for new analytic formulae. Instances of this occur in examining line bundle cohomology on surfaces and on Calabi-Yau threefolds [31][32][33][34] and in analyzing the topological invariants of knots [35][36][37][38][39][40]. Here we use the machine learning results concerning reflexive polytopes to deduce new analytic expressions for topological invariants.…”