# Exploring the Possibility of a Recovery of Physics Process Properties from a Neural Network Model

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## Abstract

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## 1. Introduction

## 2. Results

#### 2.1. The Jet Generator

#### 2.2. The Neyman–Pearson Lemma

#### 2.3. Recovering the Original Probability Distribution

#### 2.4. Calculation Results and Errors

## 3. Discussion

## 4. Materials and Methods

#### 4.1. The Jet Generator

- We start with a particle at rest with a given rest mass, here taken to be ${m}_{0}=\phantom{\rule{0.166667em}{0ex}}$100 (the units are inconsequential in the calculation).
- The particle decays into two new particles. The energies and the momenta of these particles are determined by a probability distribution. To generate the real data we use a distribution already known in particle physics, given by:$$p\left(z\right)=\mathcal{N}\frac{1+{(1-z)}^{2}}{z}\phantom{\rule{0.166667em}{0ex}}.$$The energy of the decay particle E equals $z{E}_{0}$, with ${E}_{0}={m}_{0}$ being the energy of the decaying particle. Note that the probability diverges as z approaches zero, so the distribution is limited by a lower boundary on z both due to physical and computational reasons. $\mathcal{N}$ is a constant that ensures that the integral of the probability distribution equals 1 and depends on the lower boundary set on z. In our simulation, we set the minimum z to 10${}^{-2}$, making $\mathcal{N}$ equal to $\approx 0.13$.The momentum of the decay particle is limited with the total energy of the particle. We determine the momentum by sampling the same probability distribution as for the energy, but now we set the momentum p equal to $zE$, with E being the energy of the decay particle. To differentiate between these z distributions, we write ${z}_{E}$ and ${z}_{p}$ when deemed necessary.The spatial distribution of the decay products is uniform in space. This means that, observed from the rest frame of the decaying particle, the probability that either one of the decay products flies off in a certain infinitesimal solid angle is uniform. Physically speaking, the angles $\theta $ and $\varphi $ are sampled from uniform distributions on intervals $[0,\pi ]$ and $[0,2\pi ]$, respectively.The energy, the momentum and the direction of the second particle are determined by the laws of conservation of energy and momentum. In other words, ${z}_{1}+{z}_{2}=1$ when looking at energy, and ${p}_{1}+{p}_{2}=0$, since the original momentum in the center of mass system is zero. These facts also save computational time due to symmetry, since we can sample for the energy of the first particle in the interval $\left(\right)$, instead of placing the upper limit for z to 1.
- After the first decay, the procedure repeats iteratively, i.e., we repeat step 2 for both decay products from the previous step. The only difference compared to the previous step is that we now perform the calculations for each particle in its center of mass frame and then transform the obtained quantities back to the laboratory frame, which coincides with the center of mass frame of the original particle.Once the total number of particles exceeds a pre-determined threshold (in our case set to 32), we disregard the lowest energy particles. We do this both to reduce the computational time and because we determined that these particles do not influence our end result in a significant manner.The decay procedure stops when either of two conditions is met; if the decay particle mass falls below 0.1, or a certain number of decays has been reached. In the simulations, we limited the number of decays in a single branch to 50. For simplicity, all the decays are considered to happen in the same point in space.
- The list of final decay particles now forms a list that contains the energies, the momenta and the directions of the n particles. We call this entity a jet. The jet has a maximum of 32 particles in its final state stemming from a maximum of 1 + 2 + 4 + 8 + 16 + 45·32 = 1471 decays. Hence, the full description of a jet is given by a maximum of 1471 ${z}_{E}$ parameters, 1471 ${z}_{p}$ parameters and 1471 pairs of angles ($\theta $, $\varphi $).To create the final representation of the jet which will be fed to a classifier, we create a histogram whose axes represent the direction of a particle in space. The histogram has 32 × 32 pixels with axes representing the polar angle $\theta $ and the azimuthal angle $\varphi $ of a particle. The color of a pixel in the histogram corresponds to either the energy or the momentum of the particle traveling in that direction in space. We distribute the deposited energy and momentum as Gaussian distributions in the histograms, with the Gaussian of $\sigma $ equal to 1 pixel centralized at the pixel corresponding to a direction of a certain particle. This mimics the physics situation in real life, where the readout from a detector always consists of a signal and a background noise. In fact, even when simulating data in a deterministic way, this effect is taken into account [12]. Lastly, the energy and momentum histograms are stacked to create an image with dimensions 32 × 32 × 2. An example of the jet generator tree with modified parameters is given in the appendix. Two examples of jet images are given on Figure 1 in the main body of the text.

#### 4.2. The Classifier

#### 4.3. The Algorithm Used to Recover the Underlying Probability Distributions

#### 4.3.1. Generating the Data From the Obtained Distributions

#### 4.3.2. Training the CNN Classifier

#### 4.3.3. Calculation of the Probability Distributions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## Abbreviations

QCD | Quantum Chromodynamics |

LHC | Large Hadron Collider |

RMSRE | Root mean square relative error |

CNN | Convolutional Neural Network |

AUC | Area Under the Curve |

## Appendix A. An Example of a Generated Jet

**Figure A1.**An example of the operation of the jet generator. The number on the specific node represents the total energy for a given particle, while the number on the line connecting two nodes is the energy ratio z when decaying. The decay probability distribution $p\left(z\right)$ in this image is constant. The maximum number of decays in a single branch has been set to 7, and the maximum number of particles in the jet has been set to 8. A particle stops to decay once its energy is too low (here set to 0.1). The particles coloured red are removed from the jet because their energy is too low.

## Appendix B. Supplementary Results

**Figure A2.**Several iterations of the calculated probability distributions ${p}_{p}^{i}\left(z\right)$ (symbols) compared to ${p}_{\mathrm{real}}\left(z\right)$ (full line) in the case of the guess distribution given by (7). Top: $C=0.1$. Middle: $C=10$ and Bottom: $C=100$.

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**Figure 1.**Two examples of jet images generated by the procedure outlined in the text. The x and y axes of the graphs correspond to the azimuthal angle $\varphi $ and the polar angle $\theta $ with respect to the origin. The full solid angle is mapped on these graphs, with 32 bins used for each angle. The color values in these graphs correspond to the energies of the final particles, with the energy of the original particle set to 100. The left panel shows an image of a jet generated with a probability distribution of gluon momenta radiated by a quark. The right panel shows an image of jet generated with a different probability distribution.

**Figure 2.**The left panel shows the calculated error margin vs. the iteration number in the case of the guess distribution given by (7) with C set to 10. The error calculation is described in the text. The error margins are shown separately for the case when the classifier is trained with jet images populated either with jet energies or jet momenta. The right panel shows Several iterations of the calculated probability distributions ${p}_{E}^{i}\left(z\right)$ (symbols) compared to ${p}_{\mathrm{real},E}\left(z\right)$ (full line). The 342nd iteration is the final iteration of this procedure, since the stopping condition has been satisfied.

**Figure 3.**The left panel shows the calculated error margin vs. the iteration number in the case of the guess distribution given by (7) with C set to 0.1. The right panel shows several iterations of the calculated probability distributions ${p}_{E}^{i}\left(z\right)$ (symbols) compared to ${p}_{\mathrm{real},E}\left(z\right)$ (full line). The 544th iteration is the final iteration of this procedure, since the stopping condition has been satisfied.

**Figure 4.**The left panel shows the calculated error margin vs. the iteration number in the case of the guess distribution given by (7) with C set to 100. The right panel shows several iterations of the calculated probability distributions ${p}_{E}^{i}\left(z\right)$ (symbols) compared to ${p}_{\mathrm{real},E}\left(z\right)$ (full line). The 1963rd iteration is the final iteration of this procedure, since the stopping condition has been satisfied.

**Figure 5.**The left panel shows the architecture of the convolutional neural network as described in the text. The output dimensions of each layer are given on the right side of the panel. The Blocks layer goes through 4 passes. The right panel shows the algorithm used to recover the underlying probability distributions. AUC stands for Area Under the Curve and provides an aggregate measure of the network performance.

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**MDPI and ACS Style**

Jercic, M.; Poljak, N.
Exploring the Possibility of a Recovery of Physics Process Properties from a Neural Network Model. *Entropy* **2020**, *22*, 994.
https://doi.org/10.3390/e22090994

**AMA Style**

Jercic M, Poljak N.
Exploring the Possibility of a Recovery of Physics Process Properties from a Neural Network Model. *Entropy*. 2020; 22(9):994.
https://doi.org/10.3390/e22090994

**Chicago/Turabian Style**

Jercic, Marko, and Nikola Poljak.
2020. "Exploring the Possibility of a Recovery of Physics Process Properties from a Neural Network Model" *Entropy* 22, no. 9: 994.
https://doi.org/10.3390/e22090994