Exploration of extreme QCD matter with deep learning

To study hot and dense nuclear matter, relativistic nuclear collisions are carried out experimentally, while lattice field theory provides a first-principles investigation. Meanwhile, astronomical observations of neutron stars also provide constraints on cold and dense nuclear matter. In this talk, I present the potential of deep learning based strategies to aid the exploration of QCD matter under extreme conditions, ranging from identifying essential physics from nuclear collision experiments, to facilitating lattice QCD data analysis, to efficiently exploiting astronomical observations in extracting the dense matter equation of state.


Introduction
The properties of dense and hot nuclear matter, which is governed by the strong interactions dictated from quantum chromodynamics(QCD), are unresolved, but widely studied.According to theory, a new state of deconfined matter -the quark gluon plasma (QGP) -will appear in extremely hot and/or dense conditions.On Earth, this new state of matter can be studied via heavy ion collisions (HICs) [1].At high temperature and low or vanishing baryon chemical potential, first-principle lattice QCD simulations predicted a smooth crossover transition between a dilute handronic gas and the QGP phase [2].However, at high baryon chemical potentials (and/or relatively low temperatures) lattice QCD calculations are hampered by the fermionic sign problem [3].One must therefore rely on effective models or direct experimental studies to explore the structures of the QCD phase diagram.In addition, neutron stars (NSs) provide a cosmic laboratory for the study of dense and cold neutron-rich nuclear matter.
Today, the study of extreme QCD matter is entering an advanced stage.HICs are transitioning into their precision era, as high luminosity infrastructures are planned or under construction for giving more and more high statistical measurements.In terms of lattice simulations, many of the computational barriers involved are benefiting from modern numerical algorithmic developments and also from hardware improvements, e.g., the use of GPUs.In the NSs sector, we have the discovery frontier, especially since the rapid progress in gravitational wave analysis and multi-messenger astronomy observations.Due to the complexity and large computational features involved in these sectors, the modern computational paradigm of AI methods, especially statistical learning algorithms, can naturally enter the game and provide a novel and practically powerful handle for many of the computations or difficulties [4].In the following, I will give some examples related to the extreme QCD matter research.

Decoding physics from Heavy ion collisions measurement
As the unique chance on Earth to create and study the new state of QCD matter-QGP, HICs face a big challenge in revealing the underlying physics that the collision process is a highly dynamic and superfast evolving process.Though the deconfined QGP phase may indeed be formed by the collision, it will undergo rapid expansion and cooling down, to some point its degrees of freedom will hadronize into color neutral hadronic particles, which further cascade and decay to the detector in experiment.This collision process is thus transient and impossible to resolve.What can be resolved is only the finally emitted hadrons or their decay products (e.g., their momentum and identity), and we have no direct access to the early time dynamics.Moreover, many uncertain physics factors are contained in modelling or describing the collision dynamics, which are not fully clear from theory or experiment, e.g., initial fluctuations, QCD matters' transport and bulk properties, etc.Many of these factors may entangle with multiple final observables.Consequently, it is challenging to make robust inference on the fundamental physics about the early formed QCD matter from just the limited final state measurement.
An inverse problem emerges here: given all those uncertain physics factors, well-established standard computational models, like 3+1D hyadrodynamics [5,6], can simulate the collision dynamics and get all the possible final state information.But given limited and cross-impacted final state measurements, how to decode knowledge about the early time happened physics out of the intricate entanglement influencing, this is non-trivial inverse inference task.Bayesian inference provide one possibility to tackle it [7,8].As an alternative perspective, we propose to use deep learning to capture the potentially existed inverse mapping from the final state information to the early time hot matter bulk properties.Being inspired from the success of image recognition, in [9] we adopted deep convolutional neural network (CNN) to learn this mapping directly in the sense of big data.The state of the art hydrodynamical simulations serve to generate training data with different hyperparameter setups and two types of phase transition: first order or crossover QCD transition.As shown in Fig. 1, the corresponding final state pion spectra are recorded and viewed as image-like input to the deep CNN, the network is trained to classify the involved QCD transition type which enter through the EoS in hydrodynamics.For testing data generation we tune the initial fluctuation and transport coefficients to make different and diverse collision events, also a different hydrodynamics simulation package is used compared to the training set simulation [6].Conventional observables like momentum spectrum or elliptic flow are found to hardly distinguish the two QCD transition types on the testing set.Using the deep CNN, we achieved on average 95% classification correctness on the testing events solely based on the final pion spectra.The performance is robust against the initial fluctuations and other uncertainties like the shear viscosity.As a conclusion, on hydrodynamics level for HICs, early time transition information survives to the final state, the inverse mapping is immune to other uncertainties, and with deep CNN we can approximate this decoding relationship.
We further deepened the concept into more realistic scenarios: taking into account the afterburner hadronic cascade interaction through UrQMD [10], considering the non-equilibrium transition possibility [11,12], including the experiment detector effects where hits or tracks of particles from the detector are taken as input [13], unsupervised outlier detection [14] and identifying nuclear symmetry energy in HICs [15].These demonstrated the feasibility to use deep learning in connecting experiment to theory for our exploration on extreme QCD matter.

End-to-end centrality estimation for CBM
The output of the HIC detector inherently has a point cloud structure defined as a collection of points in an unordered list with attributes e.g., position, charge, or momentum.An important property of the point cloud is that it is invariant under permutation.Accordingly, PointNet [16] is specially designed to respect this order invariance.Therefore, for HICs experiment the PointNet based models serve a way to work directly on the detector readout, and potentially allow for online event characterization.As demonstration, we discuss here the PointNet based modes for event-by-event impact parameter determination for CBM experiment.
The impact parameter b is essential for determining the event geometry and further analysis, e.g., the volume estimation in fluctuation analysis.However, in experiments we have no real control over b, which is not directly measurable.Usually, final state observable such as charged multiplicity are used to define centrality classes based on e.g., MC-Glauber model, via which only a likely distribution of b for given centrality class can be estimated.An accurate impact parameter determination on event-by-event basis is non-trivial, especially with the demand to work directly on detector output even before the particle identification.This also constructs an inverse problem.Given purely detector output (hits or tracks) for an individual event, it's non-trivial and implicit to estimate the corresponding impact parameter.
We take the supervised learning strategy similar to above QCD transition recognition to tackle this inverse problem, where we prepare training data from UrQMD followed by the CBMRoot simulation, then PointNet model is constructed to capture the inverse mapping embedded in the data [17,18].It is shown that PointNet based models can give good event-by-event impact parameter determination using the hits of charged particles and/or the tracks reconstructed from the hits, with comparison to a baseline using charged track multiplicity as input for a polynomial fit.On both the precision and accuracy sector, the PointNet based b-meter shows better performance.While the baseline model gives similar resolution (relative precision) to PointNet model in semi-central collision region, with impact parameter ranging from 3 to 16 fm, it shows a much more fluctuating and poor accuracy indicated by the mean of the prediction error in the same range.This trend becomes more evident when it comes to the realistic event distribution (∼ bdb).Considering the natural parallelizibility and fast speed, PointNet based model paves the way for real-time end-to-end event characterization.

In-medium heavy quark interaction inference from lattice QCD data
Heavy quarkonium, the bound state of the heavy quark and its anti-quark, has long been considered as a 'smoking gun' for the formation of QGP in HICs [19,20,21].Non-relativistic effective description can be allowed for heavy quarkonium due to its large mass and small relative velocity.In vacuum, the spectroscopy (masses of different states) of heavy quarkonium can be well reproduced by a Cornell-like potential.Thus, a baseline is clear to further spot medium modifications.In a thermal QCD medium, color screening will happen and weak the heavy quark interactions [22].Besides screening, the interaction can also develop a non-vanishing imaginary part manifested as thermal width inside the medium, as pointed out by the one-loop hard thermal loop (HTL) calculations [23] and the recent pNRQCD studies [24,25].
Focusing on Bottomonium (b b), to which the potential picture is more appropriate since the larger quark mass, we can use the reduced two-body Schrödinger equation to describe them, For any given complex in-medium interactions, by solving this Schrödinger equation the corresponding complex energy levels including the Bottomonium in-medium mass and thermal width can be obtained.Recently lattice QCD released the bottmonium in-medium spectroscopy for different temperatures up to 3S and 2P states [26,27], but with the currently existing one-loop HTL perturbation-based potential forms it fails in reproducing the in-medium mass and thermal width simultaneously, probably due to the missing color magnetic contribution.Accordingly, viewing this to be an inverse problem, we ask how to inversely extract one empirical effective potential from these limited spectroscopy in model independent fashion.
In Ref. [28], we devised a machine learning based approach (in left of Fig. 2), where we introduce the DNN as an unbiased and flexible enough representation to the heavy quark inmedium interaction potential.This model independent DNN representation of the potential is coupled to the Schrödinger equation solving to convert the interaction into the in-medium mass and thermal width for Bottomonium.By comparing to lattice QCD data on the mass and width, the corresponding χ 2 serves as the loss function to optimize the DNN representation, with T ∈ {0, 151, 173, 199, 251, 334} MeV and n ∈ {1S, 2S, 3S, 1P, 2P } according to the lattice QCD evaluation conditions.It's worthy to note that, the devised approach is different from conventional DNN training where the output appears in the loss directly, but rather in an implicit or physics-informed manner passing through the Schrödinger equation.With linear response analysis, we can evaluate the corresponding gradient, which arrives at the Hellmann-Feynman theorem.Then gradient descent optimization is performed to fit the DNN potential with lattice data.Furthermore, the uncertainty of the reconstructed potential is quantified by invoking Bayesian inference.Note that similar strategy is also deploed for spectral function reconstruction from correlator data [29,30].
After the verification, we applied the method on the lattice data and achieved nice agreement with the lattice QCD results for masses and thermal widths of Bottomonium simultaneously.Meanwhile, the temperature and distance dependent heavy quark potential is also obtained, as shown in the right of Fig. 2. The color screening effect clearly emerged in the reconstruction with flatter structure appearing in V R (T, r) with increasing temperature at large distance, but the temperature dependence is found to be mild compared to perturbative analysis based results in the same temperature range considered.On the other hand, the imaginary part, V I (T, r), shows significant growth both with temperature and distance, and also shows greater magnitude than one-loop HTL motivated results.

Neutron Star EoS reconstruction from M-R observationals
In the sector of studying extreme QCD matter in terms of NSs, we have explored the reconstruction of the inner EoS of NS from their astrophysical observations.NSs are one of the endpoints of stellar evolution, and the matter in the core of NS can be highly compressed, so they provide unique laboratories for studying the properties of strongly interacting dense matter.The TOV equations bridge the microscopic properties of matter, EoS, to the macroscopic massradiucs relationship of NSs, and there is one-to-one mapping between NS EoS and the NS M-R curve.So NS observables like masses and radii can help probing the underlying EoS of NSs.The recent advancement from gravitational wave observations and electromagnetic observations of NSs and their mergers results in further growing data and also an immense wave of studies in constraining the dense matter EoS.
In [31] we introduced a novel phyaics-based deep learning approach to reconstruct the NS EoS from the M-R observationals.Similar to the above inverse inference for heavy quark potential, here the proposed method involves an unsupervised learning algorithm with DNN representation for the EoS (named as EoS Network).Besides, the forward TOV solver is well emulated by a DNN in supervised manner, thus for any given EoS this pre-trained TOV-Solver Network can efficiently output the corresponding M-R curve.The EoS Network is combined with the well-trained TOV-Solver Network and optimized unsupervisedly to drive yielded M-R curve matching the NSs observational data.Within the automatic differentiation (AD) framework the gradient can be evaluated to perform the optimization in minimizing the loss function, χ 2 , with (M i , R i ) the output of the TOV-Solver Network, (M obs,i , R obs,i ) the NS M-R observations and (∆M obs,i , ∆R obs,i ) their respective uncertainties.The gradients of χ 2 with respect to the EoS Network parameters are, The proposed method is validated on several tests with mock M-R data, including also a couple of RMF EoSs, see for example Fig. 3 the reconstruction on SFHO EoS.The closure tests also showed good performance when noise is considered.Then in [32] the EoS of NS is reconstructed using the proposed DNN-based approach with the current collected NS observations.

Conclusion
The exploration of QCD matter under extreme conditions may involve complicated and challenging inverse problems, to which the modern computational methods with deep learning strategies helps from alternative and practical perspective.Two deep learning based strategies were showed in the context of QCD matter study under extreme in this proceeding, one is supervised learning by preparing training data from the forward physics modeling, then direct inverse mappings from observables to the desired physics can be learned and further transferred for prediction.The other is unsupervised learning with differentiable programming perspective for the physical modeling, where variational inference on the targeted physics parameter (or continuous function) can be efficiently performed via gradient descent, and DNN as universal approximator can be naturally integrated in the inference.

Acknowledgments
Author wish to acknowledge support from BMBF under the ErUM-Data project, the AI grant of SAMSON AG, Frankfurt and NVIDIA Corporation for donation of NVIDIA GPUs.

Figure 1 .
Figure 1.A schematic plot for QCD transition classification with HICs final particle spectra.

Figure 3 .
Figure 3. Reconstruction performance check on the SFHo EoS.