Neural ordinary differential equation and holographic quantum chromodynamics

Abstract

The neural ordinary differential equation (neural ODE) is a novel machine learning architecture whose weights are smooth functions of the continuous depth. We apply the neural ODE to holographic QCD by regarding the weight functions as a bulk metric, and train the machine with lattice QCD data of chiral condensate at finite temperature. The machine finds consistent bulk geometry at various values of temperature and discovers the emergent black hole horizon in the holographic bulk automatically. The holographic Wilson loops calculated with the emergent machine-learned bulk spacetime have consistent temperature dependence of confinement and Debye-screening behavior. In machine learning models with physically interpretable weights, the neural ODE frees us from discretization artifact leading to difficult ingenuity of hyperparameters, and improves numerical accuracy to make the model more trustworthy.

RG-Flow: A hierarchical and explainable flow model based on renormalization group and sparse prior

“…The exact holographic mapping (EHM) (Qi, 2013;Lee & Qi, 2016;You et al, 2016) further extends MERA to a bijective (unitary) flow between latent product states and visible entangled states. Recently, Li & Wang (2018); Hu et al (2020) incorporates the MERA structure and deep neural networks to design a flow-base generative model that allows machine to learn the EHM from statistical physics and quantum field theory actions. In quantum machine learning, recent development of quantum convolutional neural networks also (Cong et al, 2019) utilize the MERA structure.…”

“…The similarity between RG and deep learning has been discussed in several works (Bény, 2013;Mehta & Schwab, 2014;Bény & Osborne, 2015;Oprisa & Toth, 2017;Lin et al, 2017;Gan & Shu, 2017). The information theoretic objective that guides machine-learning RG transforms are proposed in recent works (Koch-Janusz & Ringel, 2018;Hu et al, 2020;Lenggenhager et al, 2020). The meaning of the emergent latent space has been related to quantum gravity (Swingle, 2012;Pastawski et al, 2015), which leads to the exciting development of machine learning holography (You et al, 2018;Hashimoto et al, 2018;Hashimoto, 2019;Akutagawa et al, 2020;.…”

“…Together, the latent representation of all hierarchies can be jointly inverted to generate the data. This method was recently proposed in the physics community as NeuralRG (Li & Wang, 2018;Hu et al, 2020).…”

Bulk reconstruction of metrics inside black holes by complexity

“…Nevertheless, various methods have been developed to reconstruct bulk spacetime metrics by the data of dual quantum field theories (QFTs). 1 Successful methods include the holographic renormalization [5], the reconstruction using bulk geodesics and light cones [6][7][8][9][10][11][12], the reconstruction [13][14][15][16][17][18][19][20][21][22][23][24][25] using holographic entanglement entropy [26,27], 2 the inversion formula [37] of the holographic Wilson loops [38][39][40][41], and the machine learning holography [42][43][44][45][46][47]. 3 However, all of these methods do not reconstruct the static black hole 1 Here we focus on only the bulk reconstruction of spacetime metrics.…”

“…2 Related methods include the one [28][29][30][31] using tensor networks [32,33] through the entanglement properties and the one [34] using bit threads [35,36]. 3 The holographic bulk spacetime is identified with neural networks [42][43][44][45][46][47][48][49][50][51], and the spacetimes are emergent. See [52] for a review of data science approach to string theory, and also see [53] for applications of machine learning to material sciences.…”