Simultaneous Estimation of Elliptic Flow Coefficient and Impact Parameter in Heavy-Ion Collisions using CNN

Table 1 shows the performances of all the 24 models that were built and trained for the 2 bin settings, (16 $\times$ 16) and (32 $\times$ 32). The MAE column shows the error obtained while testing the models. The percentage indicates the magnitude of the error but not the relative error. It can be observed that all 24 models performed well with maximum error being less than 6%. However, a detailed discussion predominantly focusing on the bin setting (32, 32) is given in the following section as bin (32, 32) outperformed (16, 16) in several cases.

Figure 2 shows the variation of MAE for different values of target observables and therefore compares the performance of all the 12 models under the bin setting of (32, 32). The top left panel describes the model where the output is $v_{2}$ while the top right one describes the model where the output is $b$. The bottom two panels describe the model with both outputs, $v_{2}$ and $b$ plotted separately. In each figure, the blue curve describes the model where the input was a $p_{T}$-weighted image, the red curve describes the model where the input was a mass-weighted image, the orange curve describes the model where the input was $m_{T}$ weighted image and the green curve describes the model where the inputs were both $p_{T}$ and mass-weighted images.

As evident from the figure, in predicting $v_{2}$, the CNN models performed well for $v_{2}\leq 0.4$. The MAE only increased beyond the mentioned value and can be attributed to fewer statistics in this region. But the interesting point is that despite low statistics, the CNN models could learn the pattern within a relative error ($\Delta v_{2}/v_{2}^{true}$), of 5%. Models with different inputs performed very similarly.

The prediction of $b$ is slightly different. The models seemed to perform well for mid-central regions ( (30 - 70 )%) in comparison to central and peripheral collisions. Again, this behavior could be attributed to less number of samples in these regions, thereby affecting the learning. Moreover, the worst performing class, (0-10)% centrality, only deviated by 0.25 fm with a relative error, $(\Delta b/b^{true})$, of 7.1%. In the best scenario, the relative error is as low as 0.5% which is impressive.

When it comes to inputs, the CNN models have a preference. In the case where the output was only $b$, the model with only $p_{T}$-weighted image input performed poorly while the model with only $m_{T}$-weighted input performed best. In the case where the output was both $v_{2}$ and $b$, the model with only mass-weighted image input performed poorly while the rest of the models performed more or less equally well. However, the differences in the performance was due to the contribution from low statistics regions and yet they were not significant in terms of relative error.

Figure 3 shows the centrality dependence of the target observables predicted with the model with $p_{T}$ and mass-weighted $(\eta-\phi)$ distribution as inputs. The top figure of the left panel shows the centrality dependence of $v_{2}$ and is compared with the experimental data ¹⁰ while the figure on the right shows the centrality dependence of $b$. The bottom panels show the ratio of CNN and AMPT predictions quantifying the agreement between the model and the CNN prediction.

As can be seen, the predictions of the CNN model has good agreement with the simulation data for all centrality classes. The ratio is close to one in both the cases. When compared to different (input, output) combinations, this model seems to perform well. This can be attributed to input having two layers giving maximum kinematic information. By training the CNN model with minimum bias Pb$-$Pb collisions at $\sqrt{s_{NN}}=5.02$ TeV, one allows the machine to learn physics for a larger and more complex system. This model can be directly implemented in heavy-ion collision systems tuned to LHC energies.

However, there are some discrepancies in the peripheral regions. It is to be noted that the estimation of $v_{2}$ in ALICE using $|\Delta\eta|>1$ has some level of non-flow contributions. In addition, different methods of flow estimation could also introduce a degree of uncertainty. This is the first time a single model is employed to estimate both $v_{2}$ and $b$.

Figure 4 shows the variation of $v_{2}$ with $p_{T}$ for different centrality classes. The best model for each, (16,16) and (32,32), bin is shown. Usually, fewer particles are detected for high $p_{T}$, suggesting that the input regions for high $p_{T}$ would be very sparse. Despite this constraint, all twelve models have predicted the relation in both bin settings. This feature along with the centrality dependence make these models even more robust and versatile.

In the case of convolutional neural networks (CNNs), commonly used in visual recognition tasks, various methods have been developed to visualize their behavior. One such method is class activation mapping (CAM) ²⁵, which involves global average pooling and calculating the importance of neurons within a CNN. This technique allows for the localization of crucial regions in two dimensional matrices that contribute to the model’s success in classification tasks. Building on the CAM method, Selvaraju et al. introduced a more generalized approach called gradient-weighted class activation mapping (Grad-CAM) ²⁶. Grad-CAM can be applied to a wider range of CNN architectures and provides deeper insights into the learned features of the neural network.

In the CAM method, the class activation map for class c (in the present case, c is trivial and can be suppressed because we have a regression problem instead of a classification problem) is defined as $M^{c}$.

Here, $f_{k}(A;x,y)$ represents the activation of the feature k in the final convolutional layer at spatial location $(x,y)$ of input data $A$, and $\omega^{c}_{k}$ is the weight measuring the importance of unit k for class c. In Grad-CAM, $\omega^{c}_{k}$ is defined as the result of performing global average pooling on the gradient of the score for class c for activations $f_{k}(A;x,y)$:

where $1/Z\mathbf{\sum_{x}}\mathbf{\sum_{y}}$ is the operation of global average pooling. Subsequently, the gradient-weighted class activation map is given as

The Attention Maps for the best-performing model was obtained. Figure 5 shows the Attention Maps for two samples with different input distributions. It can be observed that the region of the input space where very little information is available, demands maximum attention from the CNN. The top figure with sparsely distributed information has a bright red (high attention) heat map while the bottom plot with uniformly distributed information has a blue (low attention) heat map.

This may be surprising since CNN, or any neural networks for that matter, naturally tend to map regions of information to target variables. However, the attention maps point towards a different story. Figure 6 shows the distribution of pixel values for $p_{T}$-weighted layers across all the events. It is seen that the overwhelming majority of the pixel values lie very close to zero. This directly shows that most of the data are sparsely distributed inputs. The y-axis is log-scaled to show the sheer difference in magnitude of the number of pixels with values close to zero from those that are non-zero. The shape of the distribution is the same for all pixels. Similar kind of behavior was also seen for mass-weighted layers. This indicates that the CNN is predominantly trained on samples with highly localized regions of information. Inputs that are uniformly distributed are essentially rare and the CNN sees fewer of them while training. So, it does make sense for the CNN to effectively rely on regions of less information to extract target variables.

Neural nets usually rely on a large dataset to effectively extract mappings between input and output neurons. A sophisticated network like the one built in this work, performs well in regions of fewer statistics like for $v_{2}>0.2$ and highly central ( or peripheral) collisions. However, one can always train these models with newer datasets to further improve their accuracy. Since these models are already performing efficiently, newer datasets must exhibit good quality rather than a large quantity. Quality is subjective but it can be defined precisely for a given ML problem. In the case of weighted $\eta-\phi$ distribution as input, the quality of a dataset can be defined as the degree to which the distribution of the new dataset agrees with the one that was used for training. Any dataset that has a pixel distribution that is similar to Figure 6 is good. The CNN, when trained on such datasets, will certainly learn new patterns and will improve its performance. Deviations from the distributions of pixel values shown in Figure 6 will result in a dip in the performance of the CNN.

However, datasets from other event generators which incorporated inbuilt flow like HYDJET++ ²⁷ can also be used for training but for those datasets, the quality matters over quantity. In fact, to test the robustness of the models built in this work, one can generate data from different event generators and analyze the performance of the predictions. The $(\eta-\phi)$ distributions from the experimental data can be directly used. Since these distributions would have to most realistic distribution of pixel values, even a small dataset of 10K events should give a good understanding of how inputs must be distributed for the CNN to learn best. Such comparison would also provide some room for improvement in these Monte Carlo based event generators.