Organizations: School of Nuclear Science and Technology, University of South China, Hengyang 421001, China · INFN — Istituto Nazionale di Fisica Nucleare — Sezione di Bari, Via Orabona 4, 70125 Bari, Italy · Key Laboratory of Quark and Lepton Physics (MOE) & Institute of Particle Physics, Central China Normal University, Wuhan, Hubei 430079, China · Southern Center for Nuclear-Science Theory (SCNT), Institute of Modern Physics, Chinese Academy of Sciences, Huizhou, Guangdong 516000, China · School of Physics, East China Normal University, Shanghai 200241, China · Key Laboratory of Nuclear Physics and Ion-beam Application (MOE), Institute of Modern Physics, Fudan University, Shanghai 200433, China · Shanghai Research Center for Theoretical Nuclear Physics, NSFC and Fudan University, Shanghai 200438, China · School of Science and Engineering, The Chinese University of Hong Kong (Shenzhen), Shenzhen, Guangdong, 518172, China · School of Artificial Intelligence, The Chinese University of Hong Kong (Shenzhen), Shenzhen, Guangdong, 518172, China
Machine learning (ML) in high-energy nuclear physics (HENP) is entering a new stage in which physical knowledge is incorporated more directly into data analysis, simulation, and physics inference. This mini-review focuses on developments that have matured in the past several years. Whereas earlier applications emphasized event classification, pattern recognition, and surrogate models for selected observables, recent work has moved toward physics-integrated workflows: calibrated Bayesian extraction of QCD matter properties, dense-matter equation-of-state inference from heavy-ion and neutron-star data, generative event modeling, neural unfolding of weak physical signals, differentiable inverse solvers, gauge-equivariant and diffusion-based lattice-field samplers, and neural reconstruction of model functions in holographic QCD. We survey recent applications of ML in heavy-ion collisions, neutron-star physics, lattice QFT, and holographic or continuum QCD. The emphasis is not on ML architectures alone, but on how they enter concrete physics workflows, how physical constraints such as symmetries, conservation laws, causality, thermodynamic stability, and topology are imposed, and how uncertainty quantification and validation determine whether an AI-assisted result can support a reliable physics conclusion.
Figures & tables
Figure 1: Schematic organization of the review on machine learning in high-energy nuclear physics. The review is structured around seven interconnected themes: phenomenological workflows, Bayesian inference, experimental applications, dense-matter inference, lattice QFT, holographic and effective descriptions, and emerging foundations including foundation models, agentic scientific workflows, quantum machine learning, and benchmark infrastructure. The central hub emphasizes the integration of machine learning with physical models and data across these areas. The bottom layer highlights the common requirements for reliable physics inference—physical constraints such as symmetries, conservation laws, causality, thermodynamic stability, and topology, together with uncertainty quantification and rigorous validation.
Figure 2: Network structure of the HEIDi point-cloud generative model. A PointNet encoder, normalizing-flow latent decoder, and point-cloud diffusion module are combined to generate complete event outputs. Adapted from Ref. [ 42 ] ; see also Ref. [ 41 ] .
Figure 3: Space-time evolution of a simulated CME charge-separation pattern used in a neural-unfolding study. The figure illustrates how interactions and late-stage evolution degrade the signal that the network attempts to reconstruct. Adapted from Ref. [ 55 ] .
Figure 4: Cross-interval generalization of a symbolic-regression (SR) model for charged-hadron multiplicities. Differential multiplicities as a function of pT2 , with pT2>0.5\penalty(GeV/c)2 , are shown for several (x,Q2) intervals in the second z bin. The SR models are compared with a Tsallis fit and with the best-learned SR form f3(u)=c0/(1+c1u3) . The same equation skeleton, inferred from a subset of intervals, describes additional intervals not used during inference, illustrating the potential of SR for interpretable cross-domain generalization. Adapted from Ref. [ 62 ] .
Figure 5: Bayesian inference of the high-density nuclear EoS from heavy-ion data. The posterior bands illustrate how transport-model calculations and experimental observables constrain a density-dependent mean-field potential. Adapted from Ref. [ 79 ] .
Figure 6: Prior and posterior distributions of the mean energy loss by the quark- and gluon-initiated jets obtained for the closure tests, for the three studied distributions, obtained from marginalizing the posteriors. The color ratio is also shown. Adapted from Ref. [ 90 ] .
Figure 7: Specific shear and bulk viscosities as a function of temperature. Here we show the 60% credible interval for the full prior, the full posterior with no causality restriction, and when the initial acausal energy fraction is not allowed to exceed 10−4 . We can see in particular that bulk viscosity is significantly biased by acausal regimes [ 106 ] .
Figure 8: The GNN model extracts and encodes the topological structure of reconstructed tracks. Subsequently, the BGNN model captures the correlations among these reconstructed tracks. Finally, HF quark events are identified by an MLP model [ 121 ] .
Figure 9: Example of track reconstruction results obtained using the GNN model and projected onto the xy-plane. The green and purple lines represent true and false positive track segments, respectively, whereas the gray and red lines correspond to true and false negative track segments, respectively [ 131 ] .
Figure 10: Distributions of embedding-geometry quantities evaluated at the event level for EM and HAD showers across increasing particle multiplicity Nparticles . The top panel shows the median distance between different showers ( dintermedian ), the middle panel shows the tail of within-shower distances ( dintraQ99 ), and the bottom panel presents the separation margin ( Δ=dintermedian−dintraQ99 ). A narrow margin centered at positive or near-zero values signals a stable clustering scale, whereas broad or negative margins indicate growing ambiguity between showers. Adapted from Ref. [ 132 ] .
Figure 11: Schematic comparison of conventional holographic modeling (left) and ML-assisted holographic inference (right). In the conventional forward workflow, a specified bulk model yields observables that are compared with experimental or lattice data. In the ML-assisted workflow, selected data constrain the holographic model through a learning procedure, after which further observables are calculated. This figure is reproduced from Fig. 2 of Ref. [ 147 ] .
Figure 12: Schematic relation between holographic duality, renormalization-group flow, and deep generative architectures. The upper part illustrates the flow from the UV boundary theory to the IR through RG coarse graining, which is geometrized as motion along the AdS radial direction. The lower part shows the corresponding deep Boltzmann machine interpretation, where visible units encode boundary degrees of freedom and hidden layers represent emergent bulk scales. The figure summarizes the common intuition underlying holography, RG, tensor-network constructions, and machine-learning architectures inspired by Refs. [ 149 , 150 , 152 , 151 ] .
Figure 13: A sketch of holographic reconstruction workflow.
Figure 14: Schematic of the neural ODE framework for solving the inverse problem in QCD phase diagram analysis. Initial conditions at the black hole (BH) horizon are propagated to QCD boundary data through the equations of motion (EoMs). The neural network, initialized as a trial function for the coupling Z(ϕ) , is optimized via backpropagation to minimize the loss function L , which quantifies the deviation from lattice QCD data. The parameters ξ are updated using gradient descent. Further technical details are given in Appendix in Ref. [ 187 ] . Adapted from Ref. [ 187 ] .
Figure 15: Structured variational-autoencoder representation of neutron-star EoS ensembles. The encoder compresses an EoS into supervised stellar observables and additional latent variables. The decoder reconstructs or generates EoS curves that are then checked against physical and astrophysical constraints. Adapted from Ref. [ 210 ] .
Figure 16: Automatic-differentiation strategy for neutron-star EoS reconstruction. A neural representation of the EoS is coupled to a differentiable TOV solver so that observational residuals can be back-propagated to the EoS model. Adapted from Ref. [ 212 ] .
Figure 17: Example of field configurations generated during diffusion-model sampling of scalar lattice field theory. The panels show a denoising trajectory toward broken-phase configurations. Adapted from Ref. [ 235 ] ; see also Ref. [ 236 ] .
Figure 18: Physics-conditioned diffusion model for lattice gauge theory. Conditioning on the inverse coupling allows a single generative model to represent multiple target ensembles and to test interpolation or extrapolation across parameter values. Adapted from Ref. [ 239 ] .
Figure 19: Schematic workflow of Hamiltonian learning for quantum field theories. Experimental or quantum-simulation data are measured at finite spatial resolution, from which field correlation functions are extracted. An effective QFT Hamiltonian is then inferred by learning its relevant operator content and coupling constants. Repeating the procedure at different resolution scales provides access to scale-dependent effective Hamiltonians, connecting Hamiltonian learning with the renormalization-group picture. Adapted from Ref. [ 246 ] .
Figure 20: Agentic workflow of LQCDMaster. The planner converts natural-language LQCD tasks into a structured scientific plan; the executor turns the approved plan into PyQUDA and SLURM artifacts; expert-annotated domain skills and a deterministic Wick-contraction tool enhance reliability and efficiency. Adapted from Ref. [ 261 ] .
Figure 21: Schematic of the Schrödinger Generator for high-dimensional integration and sampling. The framework combines an adaptive coordinate transformation that reduces per-dimension variance with a normalizing-flow transformation that learns high-dimensional correlations, enabling efficient integration and low-variance sampling from complex target distributions. Upper panels illustrate the evolution of the density distribution through successive stages: uniform initial sampling (a), adaptive importance mapping (b), flow-based transformation (c), and final resampling (d). Lower panel shows the architecture of the normalizing-flow transformation based on rational quadratic splines (RQS) with a fully connected neural network (FCNN) (e). Adapted from Ref. [ 263 ] .
Figure 22: Bloch sphere: Effect of the input scaling described in f→1+1+e−w2π when applied to the ten hardest particles of a QCD and top jet with comparable jet pT . The value of w converges to a scaling factor f = 7.268 for the VQC. Adapted from Ref. [ 269 ] .
Sector
Candidate tasks
Main metrics
Key constraints
Lattice field theory
ϕ4 , U(1) , SU(2) , SU(3) sampling
Bias, autocorrelation, topology, cost
Gauge symmetry, exactness
Heavy-ion collisions
Events, spectra, flow, jets, CME/criticality
Correlations, calibration, generator shift
Conservation, acceptance
Neutron stars
EoS, mass-radius-tidal, GW/X-ray data
Coverage, phase-transition detection
Stability, causality
Functional/holographic
FRG/DSE, spectra, inverse holography
Residuals, extrapolation, uncertainty
Ward identities, UV/IR limits
Table 1: Suggested benchmark families for AI in high-energy nuclear physics. Detailed metrics are discussed in the text.
Machine learning (ML) has become integral to fundamental physics, accelerating statistical workflows from data acquisition through inference and hypothesis testing. As ML systems grow increasingly autonomous, ensuring their reliability for discovery claims becomes critical. This review synthesizes the VERaiPHY (Validation & Evaluation for Robust AI in PHYsics) initiative's frameworks for rigorous ML assessment across particle physics, astrophysics, and cosmology. We establish when verification is essential by contextualizing ML within the statistical discovery workflow. We emphasize fundamental limitations: inductive bias is unavoidable, sample complexity bounds learning, and experimental constraints limit discovery. We reflect on physicists' evolving role as both experimental designers and evaluators whose judgments encode scientific rigor into AI systems. Responsible integration requires understanding ML's transformative potential alongside its intrinsic boundaries.
Gaia Grosso, Vinicius Mikuni, Lukas Heinrich
NSF AI Institute for Artificial Intelligence and Fundamental Interactions, Cambridge, MA · MIT Laboratory for Nuclear Science, Cambridge, MA · School of Engineering and Applied Sciences, Harvard University, Cambridge, MA +3
The workflow from particle collision to physics analysis passes through a series of reconstruction steps that are traditionally modular and disconnected, with no shared representation linking low-level detector data to high-level analysis tasks. We show that casting event reconstruction as a machine learning problem naturally produces such a shared representation. We repurpose a machine learning model trained for particle-flow reconstruction (MLPF) to perform three distinct analysis tasks: jet flavor identification, jet energy regression, and missing momentum regression. By appending the per-particle latent representations learned during reconstruction as additional input features, we substantially improve over baselines that use kinematic features alone. We further demonstrate that a single linear layer trained using only the latent representations achieves competitive performance against state-of-the-art baseline architectures, and outperforms the baseline for missing momentum regression with approximately 35 times fewer parameters. These results demonstrate that the latent representations learned during reconstruction encode essential physics information needed for downstream analysis, establishing MLPF as a foundation model and offering a concrete step toward an end-to-end pipeline from detector data to physics analysis.
Farouk Mokhtar, Joosep Pata, Michael Kagan +1
University of California San Diego, La Jolla, California 92093, USA · National Institute of Chemical Physics and Biophysics, Tallinn 12618, Estonia · SLAC National Accelerator Laboratory, Menlo Park, California 94025, USA
Parton Distribution Functions (PDFs) play a central role in describing experimental data at colliders and provide insight into the structure of nucleons. As the LHC enters an era of high-precision measurements, a robust PDF determination with a reliable uncertainty quantification has become mandatory in order to match the experimental precision. The NNPDF collaboration has pioneered the use of Machine Learning (ML) techniques for PDF determinations, using Neural Networks (NNs) to parametrise the unknown PDFs in a flexible and unbiased way. The NNs are then trained on experimental data by means of stochastic gradient descent algorithms. The statistical robustness of the results is validated by extensive closure tests using synthetic data. In this work, we develop a theoretical framework based on the Neural Tangent Kernel (NTK) to analyse the training dynamics of neural networks. This approach allows us to derive, under precise assumptions, an analytical description of the neural network evolution during training, enabling a quantitative understanding of the training process. Having an analytical handle on the training dynamics allows us to clarify the role of the NN architecture and the impact of the experimental data in a transparent way. Similarly, we are able to describe the evolution of the covariance of the NN output during training, providing a quantitative description of how uncertainties are propagated from the data to the fitted function. While our results are not a substitute for PDF fitting, they do provide a powerful diagnostic tool to assess the robustness of current fitting methodologies. Beyond its relevance for particle physics phenomenology, our analysis of PDF determinations provides a testbed to apply theoretical ideas about the learning process developed in the ML community.
Amedeo Chiefa, Luigi Del Debbio, Richard Kenway
The Higgs Centre for Theoretical Physics, School of Physics and Astronomy, The University of Edinburgh, Peter Guthrie Tait Road, Edinburgh EH9 3FD, United Kingdom