Machine Learning Applied to Relativistic Heavy Ion Collision

Flow harmonics characterize the anisotropic expansion of the QGP and help in understanding the fundamental aspects of quantum chromodynamics (QCD). Evaluating flow harmonics (arXiv:2304.00336) in heavy-ion collisions is crucial because they provide insights into the properties of the quark-gluon plasma (QGP), a state of matter that existed in the early universe.

Data creation/manipulation

The Data folder is where the dataset is created. Starting from the MUSIC RHIC text files, the heat maps are created based on the transversium momentum and the azimuthal angle of all particles in a event, which each event correspond to a single image in the dataset. The "Eos_inputrework_fast.py" is how this data is transformed and "pixel_analysis.py" is a self prove of how the heat maps trully represents the original data.

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Proof I

The relationship between the number of particles in a Bin and its color value when creating the histogram.

Four bin values taken for three different events. Each matrix represents one event. The first column represents the color value of the chosen bins (normalized between 0 and 255), and the second column represents the number of corresponding particles:

$$ \begin{aligned} &\begin{bmatrix} 255 & 25 \\ 41 & 4 \\ 92 & 9 \\ 0 & 0 \end{bmatrix}, \quad \begin{bmatrix} 122 & 12 \\ 82 & 8 \\ 51 & 5 \\ 0 & 0 \end{bmatrix}, \quad \begin{bmatrix} 151 & 13 \\ 197 & 17 \\ 139 & 12 \\ 0 & 0 \end{bmatrix} \end{aligned} $$

Relationship between color value and number of particles:

$$ c = \left\lfloor \dfrac{p}{M} \cdot 255 + \dfrac{1}{2} \right\rfloor $$

where:

• ( c ): color value
• ( p ): number of particles in the bin
• ( M ): number of particles in the most filled bin

So if we take any value of the color of a certain bin and its number of particles, we can successfully estimate ( M ) by:

$$ M = \dfrac{255 \cdot p}{\lceil c - \dfrac{1}{2} \rceil} $$

For example, for event 3 we have:

$$ \begin{aligned} M_{c=151} &= 21.953 \\ M_{c=197} &= 22.005 \\ M_{c=139} &= 22.014 \end{aligned} $$

Taking the nearest integer, we get ( M = 22 ), correctly representing the number of particles in the most filled bin of the third event.

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Data validation

To validate the data provided, the YiLunDu equation of state classification paper (arXiv:1910.11530) was reproduced with 99.75% and 99.09% accuracies of train and validation datasets respectively. This reproduce was done in "Eos_network.py".

Flow estimation using Convolutional Neural Networks

"flow_idea_tt.py" and "flow_idea_ft.py" was so far the best I could achieve with regular CNNs. As far as it was possible for me to test, this method couldn't handle the task. The network couldn't return something simply better than the average of the test set. I also tried using grid search instead of regular hyperparametrization, but computational power wasnt enough to make this avaiable.

Flow estimation using Variational Autoencoders

Before going with the method choosed (CVAE, based on the results found in arXiv:1909.06296), VAE was first applied being it a simpler version so, easier and faster to apply. Unexpectedly, even though VAE did not returned a very good result, it does could find some patterns in data, since it returned some results. As far I could test it, the average R² ("accuracy") of estimating the lowest order therms of flow harmonics was 59.93% ("vae_to_fllow_3.py"). The studies continues using, finally, CVAE.

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Proof II

Regular VAE loss function composition.

Total Loss Function

$$ \mathcal{L} = \begin{pmatrix} \alpha \\ \beta \\ \gamma \end{pmatrix} \begin{pmatrix} MSE_{rec} & D_{KL} & MSE_{pred} \end{pmatrix} $$

where $$( \alpha ), ( \beta )$$, and $$( \gamma )$$ are the weights of each loss term.

Reconstruction Loss

$$ MSE_{rec} = \dfrac{1}{N} \sum_{i=1}^n (Y_i - \widehat{Y}_i)^2 $$

where $$( Y_i )$$ represents the decoder output, and $$( \widehat{Y}_i )$$ represents the input.

Kullback-Leibler Divergence

$$ D_{KL}(\mathscr{N}(\mu,\sigma^2)|\mathscr{N}(0,I)) = \dfrac{\mathcal{N}}{2} \sum_{j=1}^d (1 + \log \sigma_j^2 - \mu_j^2 - \sigma_j^2) $$

where $$( \mathcal{N} )$$ is a normalization term, and $$( \mathscr{N}(\mu,\sigma^2) )$$ is the latent space distribution, with $$( \mu )$$ as the mean and $$( \sigma )$$ as the standard deviation.

Regression Loss

$$ MSE_{pred} = \dfrac{1}{N} \sum_{i=1}^n (Y'_i - \widehat{Y}'_i)^2 $$

where $$( Y'_i )$$ is the prediction, and $$( \widehat{Y}'_i )$$ is the target.

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Flow estimation using Conditional Variational Autoencoders

Working on it ("cvae_to_flow_st.py" & "cvae_to_flow_tt.py")