Learning Particle Physics by Example: Location-Aware Generative Adversarial Networks for Physics Synthesis

Abstract

We provide a bridge between generative modeling in the Machine Learning community and simulated physical processes in high energy particle physics by applying a novel Generative Adversarial Network (GAN) architecture to the production of jet images—2D representations of energy depositions from particles interacting with a calorimeter. We propose a simple architecture, the Location-Aware Generative Adversarial Network, that learns to produce realistic radiation patterns from simulated high energy particle collisions. The pixel intensities of GAN-generated images faithfully span over many orders of magnitude and exhibit the desired low-dimensional physical properties (i.e., jet mass, n-subjettiness, etc.). We shed light on limitations, and provide a novel empirical validation of image quality and validity of GAN-produced simulations of the natural world. This work provides a base for further explorations of GANs for use in faster simulation in high energy particle physics.

Appendices

Appendix A: Additional Material

See Figs. 25, 26, 27, 28, 29, 30, 31, 32 and 33.

Fig. 25

Average signal image produced by Pythia (left) and by the GAN (right), displayed on log scale, with the difference between the two (center), displayed on linear scale

Fig. 26

Average background image produced by Pythia (left) and by the GAN (right), displayed on log scale, with the difference between the two (center), displayed on linear scale

Fig. 27

The average signal Pythia image labeled as real (left), as fake (right), and the difference between these two (middle) plotted on linear scale

Fig. 28

Average background Pythia image labeled as real (left), as fake (right), and the difference between these two (middle) plotted on linear scale

Fig. 29

Average signal GAN-generated image labeled as real (left), as fake (right), and the difference between these two (middle) plotted on linear scale

Fig. 30

Average background GAN-generated image labeled as real (left), as fake (right), and the difference between these two (middle) plotted on linear scale

Fig. 31

Difference between the average signal and the average background images labeled as real, produced by Pythia (left) and by the GAN (right), both displayed on linear scale

Fig. 32

Difference between the average signal and the average background images labeled as fake, produced by Pythia (left) and by the GAN (right), both displayed on linear scale

Fig. 33

Normalized confusion matrices showing the correlation of the predicted D output with the true physical process used to produce the images. The matrices are plotted for all images (left), Pythia images only (center) and GAN images only (right)

Appendix B: Image Pre-processing

Reference [20] contains a detailed discussion on the impact of image pre-processing and information content of the image. For example, it is shown that normalizing each image removes a significant amount of information about the jet mass. One important step that was not fully discussed is the rotational symmetry about the jet axis. It was shown in Ref. [20] that a rotation about the jet axis in $\eta -\phi$ does not preserve the jet mass, i.e. $\eta _i\mapsto \cos (\alpha )\eta _i+\sin (\alpha )\phi _i,\phi _i\mapsto \cos (\alpha )\phi _i-\sin (\alpha )\eta _i$, where $\alpha$ is the rotation angle and i runs over the constituents of the jet. One can perform a proper rotation about the x-axis (preserving the leading subjet at $\phi =0$) via

$$\begin{aligned} p_{x,i}&\mapsto p_{x,i}\end{aligned}$$

(7)

$$\begin{aligned} p_{y,i}&\mapsto p_{y,i}\cos (\beta )+p_{z,i}\sin (\beta )\end{aligned}$$

(8)

$$\begin{aligned} p_{z,i}&\mapsto p_{z,i}\cos (\beta )-p_{y,i}\sin (\beta )\end{aligned}$$

(9)

$$\begin{aligned} E_i&\mapsto E_i, \end{aligned}$$

(10)

where

$$\begin{aligned} \beta = -\text {atan}\left( \frac{p_\text {y,translated subjet 2}}{p_\text {z,translated subjet 2}}\right) - \pi /2. \end{aligned}$$

(11)

Figure 34 quantifies the information lost by various preprocessing steps, highlighting in particular the rotation step. A ROC curve is constructed to try to distinguish the preprocessed variable and the unprocessed variable. If they cannot be distinguished, then there is no loss in information. Similar plots showing the degradation in signal versus background classification performance are shown in Fig. 35. The best fully preprocessed option for all metrics is the Pix + Trans + Rotation(Cubic) + Renorm. This option uses the cubic spline interpolation from Ref. [20], but adds a small additional step that ensures that the sum of the pixel intensities is the same before and after rotation. This is the procedure that is used throughout the body of the manuscript.

Fig. 34

ROC curves quantifying the information lost about the jet mass (left) or n-subjettiness (right) after pre-processing. A preprocessing step that does not loose any information will be exactly at the random classifier line $f(x)=1/x$. Both plots use signal boosted W boson jets for illustration

Fig. 35

ROC curves for classifying signal versus background based only on the mass (left) or n-subjettiness (right). Note that in some cases, the preprocessing can actually improve discrimination (but always degrades the information content—see Fig. 34)