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  • Perspective
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Physics-driven learning for inverse problems in quantum chromodynamics

Nature Reviews Physics volume 7, pages 154–163 (2025)Cite this article

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Abstract

The integration of deep learning techniques and physics-driven designs is reforming the way we address inverse problems, in which accurate physical properties are extracted from complex observations. This is particularly relevant for quantum chromodynamics (QCD) — the theory of strong interactions — with its inherent challenges in interpreting observational data and demanding computational approaches. This Perspective highlights advances of physics-driven learning methods, focusing on predictions of physical quantities towards QCD physics and drawing connections to machine learning. Physics-driven learning can extract quantities from data more efficiently in a probabilistic framework because embedding priors can reduce the optimization effort. In the application of first-principles lattice QCD calculations and QCD physics of hadrons, neutron stars and heavy-ion collisions, we focus on learning physically relevant quantities, such as perfect actions, spectral functions, hadron interactions, equations of state and nuclear structure. We also emphasize the potential of physics-driven designs of generative models beyond QCD physics.

Key points

  • Inverse problems in physical sciences determine causes or parameters from observations.

  • Physics-driven learning integrates domain-specific physical knowledge into machine learning to solve inverse problems.

  • Physics-driven learning can help to extract physical properties and build probability distributions from data.

  • In quantum chromodynamics, physics-driven learning can deduce hadron forces, dense matter equations of state, and nuclear structure.

  • Physics-driven designs can innovate the development of deep learning and generative models.

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Fig. 1: Physics-driven designs for deep learning.
Fig. 2: A lattice gauge-equivariant convolutional neural network (L-CNN).
Fig. 3: Automatic differential framework to reconstruct hadron spectral functions from observations78.
Fig. 4: The ill-posed inverse problem from observational data to theory in the context of neutron star physics.
Fig. 5: Quantum chromodynamics (QCD) transition binary classification using convolutional neural networks (CNNs) with final particle spectra from heavy-ion collisions as input.

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Acknowledgements

The authors thank L. Brandes, Y. Fujimoto, S. Kamata, D. I. Müller, K. Murase, A. Tanaka and A. Tomiya for the helpful discussions and all collaborators for their contributions. The authors also thank the DEEP-IN working group at RIKEN-iTHEMS, the European Centre for Theoretical Studies in Nuclear Physics and Related Areas (ECT*), and the ExtreMe Matter Institute (EMMI) for their support in the preparation of this paper. G.A. is supported by STFC Consolidated Grant ST/T000813/1. K.F. is supported by Japan Society for the Promotion of Science (JSPS) KAKENHI Grant numbers 22H01216 and 22H05118. T.H. is supported by the Japan Science and Technology Agency (JST) as part of Adopting Sustainable Partnerships for Innovative Research Ecosystem (ASPIRE), grant number JPMJAP2318. S.S. acknowledges Tsinghua University under grant number 53330500923. L.W. thanks the National Natural Science Foundation of China (number 12147101) for supporting his visit to Fudan University. K.Z. is supported by the CUHK-Shenzhen University Development Fund under grant numbers UDF01003041 and UDF03003041, and Shenzhen Peacock fund under grant number 2023TC0179.

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Authors and Affiliations

  1. Department of Physics, Swansea University, Swansea, UK

    Gert Aarts

  2. Department of Physics, The University of Tokyo, Tokyo, Japan

    Kenji Fukushima

  3. Interdisciplinary Theoretical and Mathematical Sciences Program (iTHEMS), RIKEN, Wako, Japan

    Tetsuo Hatsuda & Lingxiao Wang

  4. Institute for Theoretical Physics, TU Wien, Vienna, Austria

    Andreas Ipp

  5. Department of Physics, Tsinghua University, Beijing, China

    Shuzhe Shi

  6. School of Science and Engineering, The Chinese University of Hong Kong, Shenzhen, Guangdong, China

    Kai Zhou

  7. Frankfurt Institute for Advanced Studies, Frankfurt am Main, Germany

    Kai Zhou

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The authors contributed equally to all aspects of the article.

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Correspondence to Lingxiao Wang.

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Glossary

Artificial neural networks

(ANNs). Models inspired by the structure and function of biological neural networks in human brains.

Convolutional neural networks

(CNNs). Excel with image, speech and audio inputs; they consist of three main types of layers: convolutional, pooling and fully connected layers.

Deep neural networks

Complex ANNs with multiple layers, including input, output and at least one hidden layer.

Recurrent neural networks

Bi-directional ANNs, unlike the uni-directional feedforward network; they allow outputs from nodes to influence subsequent inputs to the same nodes.

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Aarts, G., Fukushima, K., Hatsuda, T. et al. Physics-driven learning for inverse problems in quantum chromodynamics. Nat Rev Phys 7, 154–163 (2025). https://doi.org/10.1038/s42254-024-00798-x

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