A Neural Network Approach for Orienting Heavy-Ion Collision Events
A convolutional neural network-based classifier is elaborated to retrace the
initial orientation of deformed nucleus-nucleus collisions by integrating
multiple typical experimental observables. The isospin-dependent
Boltzmann-Uehling-Uhlenbeck transport model is employed to generate data for
random orientations of ultra-central uranium-uranium collisions at
$E_\text{beam} = 1\, \text{GeV/nucleon}$. Statistically, the data-driven
polarization scheme is essentially accomplished via the classifier, whose
distinct categories filter out specific orientation-biased collision events.
This will advance the deformed nucleus-based studies on nuclear symmetry
energy, neutron skin, etc.
Zu-Xing Yang, Xiao-Hua Fan, Zhi-Pan Li, Shunji Nishimura
# A Neural Network Approach for Orienting Heavy-Ion Collision Events

###### Abstract

A convolutional neural network-based classifier is elaborated to retrace the initial orientation of deformed nucleus-nucleus collisions by integrating multiple typical experimental observables. The isospin-dependent Boltzmann-Uehling-Uhlenbeck transport model is employed to generate data for random orientations of ultra-central uranium-uranium collisions at \(E_{\rm beam}=1\,\)GeV/nucleon. Given the existence of pronounced fluctuations and the impact of spectators in specific scenarios, the classifier still encounters a non-neglectable accuracy loss. Statistically, the data-driven polarization scheme is essentially accomplished via the classifier, whose distinct categories filter out specific orientation-biased collision events. This will advance the studies on nuclear symmetry energy, neutron skin, and the high-momentum tail of deformed nuclei.

## I Introduction

The multipole deformation of atomic nuclei is commonly attributed to collective motion arising from the interactions among valence nucleons [1; 2; 3; 4]. Traditionally, the quadrupole deformation can be measured experimentally from the rotational spectra of the nuclear excited state or the electric quadrupole moments derived from the hyperfine splitting of the atomic spectral line [3]. Since the year 2000, research on deformed nuclei through heavy-ion collisions has been initiated in the uranium-uranium collision system [5]. Recently, at ultra-relativistic energies, a strong linear correlation between various-order anisotropic flows and multipole deformations has been discovered [6; 7], leading to significant implications for finding signatures of the nuclear deformation.

On the other hand, with the advancement of technology, machine learning has emerged as a powerful tool for investigating various properties of nuclear structure and reactions, particularly with respect to nuclear masses [8; 9; 10; 11] and nuclear radii [12; 13; 14; 15]. Aiming at nuclear deformation, deep neural network-based generative models are being developed for producing potential energy surfaces, rotational inertia, and vibrational inertia of deformed nuclei, which can be further utilized to investigate excitation spectra within the five-dimensional collective Hamiltonian approach [16; 17]. At the same time, a series of machine learning approaches, including support vector machine, artificial neural network [18], convolutional neural network (CNN) [19], and light gradient boosting machine [20], were employed to determine impact parameters in heavy-ion collisions based on the simulation via quantum molecular dynamics (QMD) [21; 22] and ultra-relativistic QMD (UrQMD) models [23; 24; 25], achieving impact parameter identification accuracy within 0.8 fm [26; 27; 28; 29]. Subsequently, the 2D distributions of protons and neutrons on transverse momentum and rapidity plane are input to find signatures of the nuclear symmetry energy [30], providing a reliable reference for experimental studies.

From the current research perspective, an important aspect missing in the studies of deformed nuclear reactions should be noted that, while the statistical effects of deformed nuclei on heavy-ion collisions have been revealed, the orientation issue regarding individual events has not yet garnered substantial focus. Absolutely or partially orienting the collision events will undoubtedly provide a wealth of information on the reactions and structure of deformed nuclei. For example, determining the symmetry dependence of collective flows in orientations without spectators would improve the description of the equation of state, which benefits nuclear astrophysics, such as binary neutron star merger simulations [31]. Moreover, exploring higher-order deformation effects through specific orientations would yield valuable insights.

This work aims to develop a multi-input CNN to map the nuclear initial state orientations of the uranium-uranium reaction systems. Since experimental orientation information is lacking, the network will be initially trained using simulated results based on the isospin-dependent Boltzmann-Uehling-Uhlenbeck (IBUU) transport model [32; 33]. Once trained with statistical observables matching experimental data reasonably well, the network can further filter the orientations for experiments.

## II Initialization of IBUU transport model

The IBUU transport model employs Monte Carlo method to simulate the phase-space evolution of baryons and mesons during heavy-ion collisions, encompassing essential physical processes such as elastic and inelastic scattering, particle absorption, and decay [34]. The version we employed [35; 36; 37; 38; 39] has incorporated the Coulomb effect [37], Pauli blocking, and medium effects on scattering cross sections [40], etc. The used single nucleon potential includes a Skyrme-type parametrization isoscalar term and an exponential isovector term [41; 36], which
reads

\[U(\rho)=A(\rho/\rho_{0})+B(\rho/\rho_{0})^{\sigma}, \tag{1}\]

where \(\sigma=1.3\), \(A=-232\,\)MeV accounts for the attractive part, and \(B=179\,\)MeV accounts for the repulsive part. These choices correspond to an incompressibility coefficient \(K=230\,\)MeV.

As for nucleon spatial distribution, we select a relativistic mean field (RMF)-based self-consistent calculation [42; 43] with the point-coupling PC-PK1 interaction [44] including the Bardeen-Cooper-Schrieffer pairing to initialize \({}^{238}\)U. Due to its superior capability in delineating the nuclear radii, profiles, and deformations for neutron-rich nuclei, the densities generated by RMF theories have recently been utilized to initialize the IBUU model [45; 46]. Shown in Fig. 1 is the nucleon density distribution of \({}^{238}\)U with a quadrupole deformation \(\beta_{2}=0.287\).

Initially, the intrinsic coordinate systems (\(xyz\)) of the colliding nuclei are set to align with the center-of-mass coordinate system (\(x^{\prime}y^{\prime}z^{\prime}\)) of the reaction. As shown in Fig. 2, an Euler rotation operator (\(\Omega(\varphi,\theta,0)=R_{z}(\varphi)R_{y}(\theta)R_{x}(0)\)) is applied independently to the target and the projectile, where only four degrees of freedom (\(\theta_{1},\varphi_{1}\) for the target, and \(\theta_{2},\varphi_{2}\) for the projectile) are required for each collision event due to the absence of triaxial deformation. Considering that spectator fragments from the projectile can be detected on an event-by-event basis in experiments, we confine the study to the events where the projectiles are completely obstructed by the targets as

\[\theta_{2}\in\begin{cases}[\theta_{1},180^{\circ}-\theta_{1}]&\theta_{1}<90^{ \circ}\\ [180^{\circ}-\theta_{1},\theta_{1}]&\theta_{1}\geq 90^{\circ}\end{cases} \tag{2}\]

and

\[\varphi_{2}=\varphi_{1} \tag{3}\]

with \(\theta_{1}\in[0^{\circ},180^{\circ}]\) and \(\varphi_{1}\in[0^{\circ},180^{\circ}]\). In experiments, events with the target spectator fragments appearing only in one direction can also be filtered out to determine ultra-central collisions. As a comparison, nucleon density distribution constrainingly calculated at \(\beta_{2}=0\) point are also exploited to initialize the spatial distribution. Under the present densities, the local density approximate Fermi gas method is employed to generate the momentum distributions. Within the intermediate energy range, increasing the beam energy enhances the deformation effects [45], and also produces more hadrons within a single event, which is advantageous to trace back the orientation. Accordingly, the beam energy is adjusted to \(1\,\)GeV/nucleon.

## III Convolutional orientation filter

In the IBUU transport model, the comprehensive information of the final state is stored in the phase-space distribution \(f(\vec{r},\vec{p},T)\), where \(\vec{r}\) and \(\vec{p}\) represent the coordinate and momentum of a hadron, respectively. The variable \(T\) denotes the type of the hadron, which includes \(p\), \(n\), \(\pi^{+}\), \(\pi^{0}\), \(\pi^{-}\), \(\Delta^{++}\), \(\Delta^{+}\), \(\Delta^{0}\), \(\Delta^{-}\). As reactions occur, the anisotropy in coordinate space is transferred to momentum space. As such, the redundant information in the phase-space can be eliminated, retaining only the physical quantities that are empirically considered relevant to the collision orientation and organized into three

Figure 2: Simulated schematic for uranium-uranium collision, where \(x-z\) plane is the reaction plane.

Figure 1: The ground-state proton (left panel) and neutron (right panel) densities for \({}^{238}\)U in the \(r_{\perp}-x\) (\(r_{\perp}=\sqrt{(y^{2}+z^{2})}\)) coordinate system calculated by relativistic mean field with PC-PK1 interaction.
aspects as follows:

1. The hadron counting \(C(p_{t},y_{0})\) and distributions of anisotropic flows \(v_{n=1,2,3,4}(p_{t},y_{0})\) in central rapidity \(y_{0}(=y/y_{\rm beam})\) and transverse momentum \(p_{t}\) (\(=\sqrt{p_{x}^{2}+p_{y}^{2}}\)). In practice, we adopted grid intervals of \(\Delta p_{t}=0.15\,\)GeV/c and \(\Delta y_{0}=0.1\) for transverse momentum and center rapidity bins, respectively.
2. Mean values of anisotropic flows \(\langle v_{n}\rangle\,(n=1,2,3,4)\) for all the emitted particles.
3. Counting of the high-momentum nucleons (\(p_{t}>0.5\,\)GeV/c) in transverse emission azimuth angle \(\phi\) (\(=\arccos\left(p_{x}/p_{t}\right)\)) and longitudinal emission angle \(\theta_{\perp}\) (\(=\arccos\left(p_{z}/|\vec{p}|\right)\)). The angular intervals are uniformly set at \(\Delta\theta_{\perp}(\Delta\phi)=30^{\circ}\).

In particular, the anisotropic flows are expressed as

\[v_{n}=\left\langle\cos(n\phi)\right\rangle. \tag{4}\]

The organized observables will serve as inputs for the network, as depicted in Fig. 3. Hereinafter, the network will be denoted as the convolutional orientation filter (COF). The output was anticipated to include \(\theta_{1}\), \(\theta_{2}\), \(\varphi_{1}\), and \(\varphi_{2}\), nevertheless, after multiple rounds of training and validation, we have arrived at the following conclusions: 1. According to the Heisenberg uncertainty principle, considering this issue as a linear regression task is relatively less reliable compared to adopting a classification approach; 2. The angle \(\varphi\) demonstrates limited sensitivity to the present inputs; 3. \(\theta=k\) and \(\theta=180^{\circ}-k\) are degenerate states and cannot be distinguished. As such, we categorize the output, based on the \(\theta_{1}\) and \(\theta_{2}\), into six cases (as illustrated in Table 1) and perform one-hot encoding for representation. In this case, a tip-body collision (\(\theta_{1}=0^{\circ},\theta_{2}=90^{\circ}\)) would be labeled as \([0,0,1,0,0,0]\).

The robustness of final-state harmonic flow information within a single event is significantly influenced by the number of emitted particles. At relativistic energies, the production of hadrons is often increased by orders of magnitude compared to intermediate energies, making the identification of the initial state orientation markedly more reliable. To provide a preliminary estimation, we temporarily neglect the stronger shadow effect originating from spectator fragments under intermediate energy reactions and consider an average of 20 events with identical initial orientations for comparative purposes.

The three aforementioned inputs are individually fed into multiple branch cells composed of convolutional, pooling, and fully connected layers. The outputs of these branch cells are then aggregated in a trunk cell and subjected to batch normalization [47], ensuring a stable feature distribution. An essential point that must be emphasized is that the network consists of 15 layers with intricate connections, whose complexity can lead to some neurons being trapped in the negative range and becoming deactivated under the commonly used ReLU

\begin{table}
\begin{tabular}{c|c c} \hline \hline Case & \(\theta_{1}\) (Target) & \(\theta_{2}\) (Projectile) \\ \hline
1 & \([0^{\circ},30^{\circ}]\cup[150^{\circ},180^{\circ}]\) & \([0^{\circ},30^{\circ}]\cup[150^{\circ},180^{\circ}]\) \\ \hline
2 & \([0^{\circ},30^{\circ}]\cup[150^{\circ},180^{\circ}]\) & \([30^{\circ},60^{\circ}]\cup[120^{\circ},150^{\circ}]\) \\ \hline
3 & \([0^{\circ},30^{\circ}]\cup[150^{\circ},180^{\circ}]\) & \([60^{\circ},120^{\circ}]\) \\ \hline
4 & \([30^{\circ},60^{\circ}]\cup[120^{\circ},150^{\circ}]\) & \([30^{\circ},60^{\circ}]\cup[120^{\circ},150^{\circ}]\) \\ \hline
5 & \([30^{\circ},60^{\circ}]\cup[120^{\circ},150^{\circ}]\) & \([60^{\circ},120^{\circ}]\) \\ \hline
6 & \([60^{\circ},120^{\circ}]\) & \([60^{\circ},120^{\circ}]\) \\ \hline \hline \end{tabular}
\end{table}
Table 1: The initial collision orientations corresponding to the six classification cases.

Figure 3: Schematic diagram of the structure of the convolutional orientation filter (COF) neural network.
[48] activation function for nonlinearity, especially after multiple iterations. To address this issue and improve convergence, we adopt the LeakyReLU activation function [49] to avoid gradient vanishing. Ultimately, through multiple iterations constraint via the cross-entropy loss function [50], the network generates orientation classifications that have been normalized by the Softmax function [51]. The meticulously designed hyperparameter set for the COF network architecture is listed in Appendix A, which can also be employed for other transport models, such as a multiphase transport model [52; 53; 54], UrQMD model [23; 24; 25], and antisymmetrized molecular dynamics model [55; 56], etc.

## IV Results and Analysis

In the classic picture, the orientation of collisions is random, necessitating averaging over all events to search for the signatures of deformation. Before discussing the COF generation, the observable \(\langle\langle v_{2}\rangle_{l}^{2}\rangle\) needs to be examined, as it has been proven to be a sensitive probe of quadrupole deformation [7]. The \(\langle\langle v_{2}\rangle_{l}^{2}\rangle\) is shown (the blue color series) in Fig. 4, where the inner and outer brackets respectively represent the averaging elliptic flow over \(l\) events with the same orientation and averaging over all orientations. The response relation between \(\langle\langle v_{2}\rangle_{l}^{2}\rangle\) and the quadrupole deformation \(\beta\) can be expressed as

\[\langle\langle v_{2}\rangle_{l}^{2}\rangle=a_{l}+f(\beta), \tag{5}\]

where the fluctuation \(a_{l}=\langle\langle v_{2}\rangle_{l}^{2}\rangle_{\beta=0}\) corresponds to the blue dashed curve and the deformation effect \(f(\beta)\) is manifested as the difference between the blue solid and dashed curves. Only \(l=1\) is experimental observable, while larger \(l\) characterizes the variations in the strength of deformation effects as the hadron yield increases in reaction systems. In this case, we define the strength of deformation effect as

\[I=\frac{f(\beta)}{a_{l}}, \tag{6}\]

which corresponds to the red curve of Fig. 4. As \(l\) increases, the number of produced hadrons in the same orientation increases, leading to a reduced fluctuation, while the \(f(\beta)\) remains constant during this process, resulting in increasing strength of deformation effect. The constant \(f(\beta)\) also indirectly indicates that the geometric effect is largely independent of the baryon number of a reaction system.

We shift our attention to the predictions of the COF network. Due to the abundance of simulated events for training, overfitting has been effectively mitigated, indicating that the results generated by the network exhibit strong generalization ability. The final analysis reveals that when combining 20 eve

Figure 4: The mean square elliptic flows resulting from the combination of \(l\) events with the same orientation (curves in the blue color series) and the strength of the deformation effect \(I\) (curves in the red color series). See text for details.

Figure 5: The probability distribution of the actual case at different predicted values (channel) with (a) combining 20 events \(l=20\) and (b) individual events \(l=1\).
tion for training, the accuracy reaches an impressive 67%, while in the case of event-by-event training, the accuracy dramatically drops to a mere 28%. The accuracy is positively correlated with \(I\), but even for \(l=20\), the deformation-induced effects remain comparable to the fluctuations (see Fig. 4). While future improvements might enhance COF performance, there will inherently remain unclassifiable events due to statistical fluctuation errors.

Displayed in Fig. 5 are the probability distributions of the actual cases at different predicted cases, where the normalization condition is \(\sum_{i(\text{or }j)}P_{i(\text{or }j)}=1\) with the subscript \(i,j\) representing the predicted and the true cases, respectively. Taking the black solid line in (a) as an example, the situation classified as case-1 is actually composed of \(\sim 80\%\) case-1 and \(\sim 20\%\) case-2. To avoid confusion and consider the statistical significance of each prediction, we will designate the six predictions as six channels. It is evident that channel-1 and -6 exhibit better discriminative power, which corresponds to the commonly mentioned body-body and tip-tip collisions [5; 7] and are crucial for investigating various properties of deformed nuclei. For Fig. 5(b), the network polarization performance explicitly diminishes, yet the partial polarization brought by channel-1 and -6 remains effective. It is worth emphasizing that, due to the limited capability of the IBUU model to accurately emulate spectator fragments, the capability of COF to distinguish between cases with only trace spectator fragments (case-1, -4, and -6) and cases with a larger number of spectator fragments (case-2, -3, and -5) may also be weakened. The spectator fragments are experimentally distinguishable on an event-by-event basis, we hence renormalize case-1, -4, and -6 in channel-1, obtaining the proportion of body-body collisions is 45% with \(l=1\).

Empirically, elliptic flow \(v_{2}\) is sensitive to collision orientations, the count proportion distributions of which across different channels are displayed in Fig. 6. The six channels are observed to be statistically divided into three categories in Fig. 6(a) with \(l=20\). Channel-1 exhibits a higher distribution in regions with stronger \(v_{2}\), channel-2 and -4, as the second tier, progressively cluster around the zero elliptic flow intensity, and channel-3, -5, and -6 encompass the category of events with the weakest elliptic flow. This order is clearly related to the initial state overlap eccentricity, where the overlap refers to the projection of the participants on the \(x-y\) plane [34]. Eventually, we discover that these three categories appear to be independent of the target orientation and are solely determined by the projectile orientation. Turning to the case of \(l=1\) as shown in (b), the corresponding patterns are largely attenuated as anticipated due to fluctuations. Statistical regularities also emphasize a key point that it is challenging to distinguish the orientations of colliding nuclei through conventional data processing methods since the appreciable amount of events with zero \(v_{2}\) regardless of channel. As such, the emerging artificial intelligence technologies geared toward combining multiple observables may be the only solution to this problem.

Finally, the mean squared elliptic flows \(\langle v_{2}^{2}\rangle\) across different channels are investigated and shown in Fig. 7. The \(\langle v_{2}^{2}\rangle\) for different orientation cases has been theoretically calculated and is represented by the blue dashed line, which can also be interpreted as the distribution predicted by the network with perfect accuracy. For the case of (a) \(l=20\), the \(\langle v_{2}^{2}\rangle\) of the screened events incredibly aligns exactly with the theoretical calculation, affirming the accuracy of COF reaches a level where it is no longer sensitive to the observable. Adhering to the Heisenberg uncertainty principle, this suggests that further improvement in accuracy is difficult. Compared to the average over all events (red dashed line), channel-1 vividly amplifies the effect of deformation, which corresponds virtually to the results shown in Fig. 6 (a). For (b) \(l=1\), a substantial disparity between the COF channels and the theoretical ideal cases occurs, which can be attributable to the non-constant fluctuations in different channels caused by the strong individual fluctuation in the training set. Hence, the fluctuation patterns need to be reconsidered in an orientation-dependent scheme as

\[\langle v_{2}^{2}\rangle_{\text{channel,Theo-2}}=a_{\text{channel}}+f(\beta)_{ \text{channel}} \tag{7}\]

Figure 6: The distributions of event count proportions in the elliptic flow across different predicted values (channels) with (a) combining 20 events \(l=20\) and (b) individual events \(l=1\).
with

\[f(\beta)_{\rm channel}=\langle v_{2}^{2}\rangle_{\rm channel,Theo-1}-a_{\rm avg,}, \tag{8}\]

where the fluctuation \(a_{\rm channel}=\langle v_{2}^{2}\rangle_{\rm channel,\beta=0}\) corresponds to the collisions between spherical nuclei, and the \(a_{\rm avg.}\) is the average value for all events (red parallel line). In particular, the \(a_{\rm channel}\) can be measured experimentally by selecting a nearby spherical nucleus, such as \({}^{208}\)Pb. The \(\langle v_{2}^{2}\rangle_{\rm channel}\) from the above calculation is represented by the magenta dash-dot line labeled as Theo-2. It can be seen that for channel-1, -4, and -6, which only involve a limited number of spectators, the difference between COF and the Theo-2 results is relatively minor, ascribed to the circumstance that when calculating fluctuations through ultra-central spherical nucleus collisions, spectators are entirely absent. This demonstrates that spectators play an important role in elliptic flow at current energy, deserving a more in-depth study. From another perspective, for the real scenarios represented by case-1, -4, and -6, it is feasible to accurately measure the response relation between initial eccentricity and elliptic flow.

With the aid of the COF network, orientation-dependent physical quantities would become experimentally measurable, which will enable us to capture more information from laboratories such as HIAF/IMP, J-PARC, FAIR, NICA, HIMAC, etc., and will greatly deepen our understanding of the geometric effects in heavy-ion collisions. In future research, utilizing the elliptic flow in channel-1 might offer a geometric effect-based means to further explore the impact of symmetry energy and equation of state. Furthermore, theoretical research [57] has indicated that high-momentum tails, resulting from short-range correlations, are more likely to manifest at the nuclear surface. Therefore deformed nuclei with larger surface areas will undoubtedly provide an excellent platform for investigating this phenomenon, which will also benefit from the COF network. In summary, this research would greatly broaden the horizons of heavy-ion collision studies, facilitating a deeper comprehension of both nuclear structure and nuclear reactions.

## V Summary

Based on the 1 GeV/nucleon uranium-uranium ultra-central collision data simulated using the IBUU transport model, we constructed a composite neural network of multiple observables in a supervised classification mode to retrospectively determine the initial state orientation of the deformed colliding nuclei. Although non-negligible errors persist, this data-driven polarization approach is applicable in a statistical sense. The different channels of the COF network filter out specific orientation-biased collision events. Evidently, the identification capability of the neural network is fundamentally positively correlated with the ratio of deformation effect to fluctuation, where the fluctuation is related to the number of final-state particles and the deformation effect can be diminished to some extent by the influence of spectators at 1 GeV/nucleon. We hence assert that the COF network will certainly be more effective in high-energy reactions.

By contrasting the elliptic flow distributions between the ideal and COF conditions, we observe that spectators indeed play a significant role in reactions, thereby also influencing the determination of orientations. Combining various deformation constraint points (which can be replaced in experiments by different reaction systems, such as Pb-Pb and U-U collision systems), it becomes possible to accurately measure the response relation between the initial eccentricity and the mean square elliptic flow at \(\theta_{1}\approx\theta_{2}\). More importantly, analyzing other observables for the selected events from one or several COF channels will enhance our understanding of geometric effects and aid us in utilizing these effects to explore other nuclear properties, such as the equation of state, short-range correlations, and neutron skin thickness. We anticipate the current network to be combined with a broader range of

Figure 7: The distribution of mean squared elliptic flow \(\langle v_{2}^{2}\rangle\) across different predicted values (channels) with (a) combining 20 events \(l=20\) and (b) individual events \(l=1\). The blue dashed lines represent the ideal scenario when perfect classification is achieved, referred to as Theo-1. The magenta dotted line represents the ideal scenario with an orientation-dependent fluctuation caused by the network, labeled as Theo-2. The red parallel lines represent the average value for all events.
transport models and applied to experimental data measurements as soon as possible.

## VI Acknowledgements

We acknowledge helpful discussions with Prof. Haozhao Liang. This work is supported by the National Natural Science Foundation of China under Grants Nos. 12005175, 11875225. the Fundamental Research Funds for the Central Universities under Grant No. SWU119076, the JSPS Grant-in-Aid for Early-Career Scientists under Grant No. 18K13549, the JSPS Grant-in-Aid for Scientific Research (S) under Grant No. 20H05648. This work was also partially supported by the RIKEN Pioneering Project: Evolution of Matter in the Universe and the INT Program INT-23-1a and Institute for Nuclear Theory.

## Appendix A The Hyperparameter Set of Convolutional Orientation Filter

As indicated in Table 2, the COF network consists of 6 cells, denoted as Cell-n (n=1-6), among which, except for cell-1, which is a convolutional neural network (CNN), all other cells are fully connected neural networks (FCNN). Each cell is composed of several unit layers, including convolutional layer (Conv.), average pooling layer (pooling), fully connected layer (Linear), and batch normalization layer (Batch-Norm).

In the presented table, the initial row and the concluding row of each cell respectively signify its input and output. In this context, \(\text{Input}_{1}\) comprises the following five matrices: \(C(p_{t},y_{0})\), \(v_{1}(p_{t},y_{0})\), \(v_{2}(p_{t},y_{0})\), \(v_{3}(p_{t},y_{0})\), \(v_{4}(p_{t},y_{0})\). \(\text{Input}_{2}\) denotes the mean values of anisotropic flows \(\langle v_{n}\rangle\) (where \(n=1,2,3,4\)) across all emitted particles. The \(\text{Input}_{3}\) and \(\text{Input}_{4}\) correspond to the hadron count in transverse and longitudinal emission azimuth angle. The input for Cell-5 is a combination of the outputs from Cell-3 and Cell-4 noted as \(\text{Output}_{3}\uplus\text{Output}_{4}\), where the symbol \(\uplus\) indicates splicing two vectors, e.g., \([1,3,2]\uplus[2,5,7]=[1,3,2,2,5,7]\). Similarly, the input for Cell-6 is a combination of the outputs from Cell-1, Cell-2, and Cell-5.

For the header of Table 2, "\(D\)" represents the output dimension of the layer, "\(C_{\text{in}}\)" denotes the input channels, "\(C_{\text{out}}\)" signifies the output channels, "\(K_{s}\)" refers to the size of the kernel which includes both convolutional and pooling dimensions, "\(S_{t}\)" indicates the stride used during convolution or pooling operations, and "\(g(x)\)" represents the non-linear activation function with LReLU is \(\text{LeakyReLU}(x)=\max\{0.01\times x,x\}\).

Furthermore, to mitigate the influence of absolute data magnitudes, all input features are subject to min-max normalization, ensuring they fall within the range of 0 to 1. During the training process, a dynamically decreasing learning rate, which reduces as the loss function converges, is applied with the Adaptive Momentum Estimation (Adam) [58] optimizer. The aforementioned definitions and concepts are entirely consistent with the patterns used in PyTorch [59].
