Machine learning action parameters in lattice quantum chromodynamics

Numerical lattice quantum chromodynamics studies of the strong interaction are important in many aspects of particle and nuclear physics. Such studies require significant computing resources to undertake. A number of proposed methods promise improved efficiency of lattice calculations, and access to regions of parameter space that are currently computationally intractable, via multi-scale action-matching approaches that necessitate parametric regression of generated lattice datasets. The applicability of machine learning to this regression task is investigated, with deep neural networks found to provide an efficient solution even in cases where approaches such as principal component analysis fail. The high information content and complex symmetries inherent in lattice QCD datasets require custom neural network layers to be introduced and present opportunities for further development.

Figure 1

Contours show the scale setting quantities $t_0$ and ${\omega }_0$, as well as the lattice spacing times the pion mass $a{m}_{\pi }$, and rho meson mass $a{m}_{\rho }$, determined using calculations on each ensemble in the two $L/a=12$ grids. The stars show the locations of the ensembles from Grids A (blue) and B (orange).

Figure 2

Diagrammatic representation of the construction of planar Wilson loops $W_{k\times l}(x)$, with indices $k$ and $l$ denoting the dimensions of the loop (with orientation label suppressed), from gauge links $U_{\mu }(x)$.

Figure 3

Autocorrelation functions $\rho (\tau )/\rho (0)$ [left, defined in Eq. (5)] and autocorrelation times ${\tau }_{\mathrm{int}}$ [right, defined in Eq. (6)] of the pion (top) and $\rho$ (center) two-point correlation functions at different Euclidean time separations, and of the various space-time averaged $n\times m$ planar Wilson loops (bottom). Measurements are performed on a subset of ensemble F1, for $N_{\mathrm{traj}}=4000$ sequential trajectories ($N_{\mathrm{traj}}=7980$ for the loops). The colors identify the type of loop and the shaded bands correspond to the uncertainties on these quantities as determined from a bootstrap procedure using $N_{\mathrm{boot}}=100$ bootstrap resamplings of size $N_{\mathrm{traj}}$.

Figure 4

Contours show $\ln |W_{n\times m}|$ from evaluations on each ensemble in the two $L/a=12$ grids. The stars show the locations of the ensembles from Grids A (blue) and B (orange).

Figure 5

Eigenvalues $e_n$ (left panel) and eigenvectors $v_n$ (right panel) of the loop correlation matrix for Grid A. The strength of the contribution of each loop to each eigenvector is represented by the tone of the corresponding box in the right panel (i.e., $\mathrm{darker}=\mathrm{larger}\mathrm{contribution}$).

Figure 6

Combinations of loops corresponding to the four largest eigenvectors of the loop correlation matrix for Grid A. Each color denotes a different ensemble in Grid A.

Figure 7

The Jensen-Shannon divergence, $D_{\mathrm{JS}}$, between pairs of ensembles in Grid A, calculated over the three-dimensional distributions defined by the three dominant eigenvectors of the loop correlation matrix used for the PCA. $D_{\mathrm{JS}}=1$ implies completely distinguishable distributions.

Figure 8

Combinations of loops corresponding to the dominant eigenvectors of the loop correlation matrix for ensemble Sets D and E. Each color denotes a different ensemble.

Figure 9

The Jensen-Shannon divergence, $D_{\mathrm{JS}}$, between pairs of ensembles in Sets D (left) and E (right), calculated over the three-dimensional distributions defined by the three dominant eigenvectors of the loop correlation matrix used for the PCA. The maximum value of $D_{\mathrm{JS}}$ in each Set is 0.6. $D_{\mathrm{JS}}=1$ corresponds to completely distinguishable distributions.

Figure 10

A schematic representation of the neural network structure used for parametric regression. Gauge links, expressed in an SU(2) basis as 4 real numbers, are used as inputs to the network. There are 4 links in each positive direction from a given site, giving a total of $4\times 4\times V=16\times 12^3\times 36=995328$ real numbers per gauge field. Two fully connected layers, each with 96 nodes, are used. Each hidden layer features a tanh activation function and dropout. A random set of connections between layers are omitted to denote dropout.

Figure 11

Predictions of $\beta$ and $m_0$ on validation ensembles at the same parameter values as the training ensembles. The stars in the left panel denote the parameters used to generate the ensembles, while the ellipses show the one-standard deviation confidence interval of the predictions for the validation ensembles. The same validation data are shown as histograms in the right figure, with the intersections of the grid lines indicating the parameters used for ensemble generation.

Figure 12

Autocorrelation function in Monte-Carlo time [left, defined in Eq. (10)] and autocorrelation time [right, defined in Eq. (6)] of the feature distinguishing two streams at the same set of parameters, trained on sequences of gauge field configurations. The autocorrelation function was generated by averaging over many different results (trained using all different pairs of the 10 streams, $F_{1,\dots ,10}$, at the same parameters), and was found to be robust under changes of the network structure used to generate it. The dashed horizontal line on the right figure shows the maximum autocorrelation time of various physics observables (see Fig. 3).

Figure 13

Diagrams of the neural network structure used. In the first layer, SL, $CP$, or SLCP structures are formed, e.g., in the $CP$ case, products of the 18 different types of loops separated by lattice distance $R<13$ (averaged in integer space bins of $R$) are allowed, for a total of $18\times 18\times 13=4212$ loop products. The first layer is followed by 3 fully connected hidden layers with 1024, 512, and 256 nodes. Each hidden layer uses a tanh activation function, with dropouts between layers.

Figure 14

Predictions of $\beta$ and $m_0$ for the validation ensembles in Grid A at the same parameter values of the training ensembles, using SL (left panel), SLCP (right panel) and $CP$ (bottom panel) network structures. The stars show the location of each ensemble in parameter space, while the ellipses show the $1\sigma$ confidence regions generated from the variation of the predictions for the 100 validation samples from each ensemble.

Figure 15

Loss for networks trained on the ensembles in Grid A with SL (orange), $CP$ (blue), and SLCP (green) structures in the first layer, optimized with the Adam optimizer. The dark lines indicate the training loss and the pale lines show loss on the validation data.

Figure 16

Predictions of $\beta$ and $m_0$ from the $CP$ network trained on Grid A, for the ensembles in Grid B (left panel) and Sets D and E (right panel). The open circles show the location of each ensemble in parameter space, while the ellipses show the $1\sigma$ confidence regions generated from the variation of the predictions for the 100 validation samples from each ensemble. The greyed-out stars and ellipses show the validation data and training ensemble locations.

Figure 17

Predictions of $\beta$ and $m_0$ from a $CP$ network trained on a subset of the ensembles in Grid A. The stars show the location of each ensemble in parameter space, while the ellipses show the $1\sigma$ confidence regions generated from the variation of the predictions for the 100 validation samples from each ensemble. In the right panel, the open circles show the location of testing ensembles, that were not included in training, in the parameter space, while the matched-color ellipses show the $1\sigma$ confidence regions of the network predictions.

Figure 18

Predictions of $\beta$ and $m_0$ for the validation ensembles in Grid C at the same parameter values of the training ensembles, using a $CP$ network structure. The stars show the location of each ensemble in parameter space, while the ellipses show the $1\sigma$ confidence regions generated from the variation of the predictions for the 100 validation samples from each ensemble.

Figure 19

The various Wilson loops, $W_{m\times n}$, on the first 1600 (of 10000) trajectories of each of the ensembles in Grid A (left column), Grid B (middle column) and Grid C (right column).

Figure 20

The various Wilson loops, $W_{m\times n}$, on each of the ensembles in Grid A for all $m$, $n$ combinations used in this work.

Figure 21

The various Wilson loop correlators, ${\mathcal{W}}_{m\times n,p\times q}(r)$, on each of the ensembles in Grid A for a selection of choices of loop shapes and separations $r=0$, 1.

Figure 22

The various Wilson loops, $W_{m\times n}$, on each of the ensembles in Grid B for all $m$, $n$ combinations used in this work.

Figure 23

The various Wilson loop correlators, ${\mathcal{W}}_{m\times n,p\times q}(r)$, on each of the ensembles in Grid B for a selection of choices of loop shapes and separations $r=0$, 1.

Figure 24

The various Wilson loops, $W_{m\times n}$, on each of the ensembles in Grid C for all $m$, $n$ combinations used in this work.

Figure 25

The various Wilson loop correlators, ${\mathcal{W}}_{m\times n,p\times q}(r)$, on each of the ensembles in Grid C for a selection of choices of loop shapes and separations $r=0$, 1.