184 lines
15 KiB
BibTeX
184 lines
15 KiB
BibTeX
@article{enn-review-kon25,
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title = {The principles behind equivariant neural networks for physics and chemistry},
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volume = {122},
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issn = {0027-8424},
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url = {https://www.ncbi.nlm.nih.gov/pmc/articles/PMC12541325/},
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doi = {10.1073/pnas.2415656122},
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abstract = {A distinguishing feature of the neural network models used in Physics and Chemistry is that they must obey basic underlying symmetries, such as symmetry to translations, rotations, and the exchange of identical particles. Over the course of the last several years, the artificial neural networks community has developed a class of networks called group-equivariant neural nets that can efficiently “bake-in” such symmetries into the structure of the network itself. Equivariant neural nets leverage ideas from group representation theory and express all variables in the generalized Fourier space corresponding to the underlying group. In this article, we review this formalism and derive the general form of operations allowable in equivariant neural networks. Specifically, we discuss why the Clebsch–Gordan transform appears in such architectures, and how it can play the role of an equivariant nonlinearity.},
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number = {41},
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urldate = {2026-05-04},
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year = {2025},
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journal = {Proceedings of the National Academy of Sciences of the United States of America},
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author = {Kondor, Risi},
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pmid = {41052329},
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pmcid = {PMC12541325},
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pages = {e2415656122},
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}
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@misc{enn-review-fc26,
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title = {Equivariant {Neural} {Networks} for {Force}-{Field} {Models} of {Lattice} {Systems}},
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url = {http://arxiv.org/abs/2601.04104},
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doi = {10.48550/arXiv.2601.04104},
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abstract = {Machine-learning (ML) force fields enable large-scale simulations with near-first-principles accuracy at substantially reduced computational cost. Recent work has extended ML force-field approaches to adiabatic dynamical simulations of condensed-matter lattice models with coupled electronic and structural or magnetic degrees of freedom. However, most existing formulations rely on hand-crafted, symmetry-aware descriptors, whose construction is often system-specific and can hinder generality and transferability across different lattice Hamiltonians. Here we introduce a symmetry-preserving framework based on equivariant neural networks (ENNs) that provides a general, data-driven mapping from local configurations of dynamical variables to the associated on-site forces in a lattice Hamiltonian. In contrast to ENN architectures developed for molecular systems -- where continuous Euclidean symmetries dominate -- our approach aims to embed the discrete point-group and internal symmetries intrinsic to lattice models directly into the neural-network representation of the force field. As a proof of principle, we construct an ENN-based force-field model for the adiabatic dynamics of the Holstein Hamiltonian on a square lattice, a canonical system for electron-lattice physics. The resulting ML-enabled large-scale dynamical simulations faithfully capture mesoscale evolution of the symmetry-breaking phase, illustrating the utility of lattice-equivariant architectures for linking microscopic electronic processes to emergent dynamical behavior in condensed-matter lattice systems.},
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urldate = {2026-05-04},
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publisher = {arXiv},
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author = {Fan, Yunhao and Chern, Gia-Wei},
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month = jan,
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year = {2026},
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note = {arXiv:2601.04104
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version: 1},
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keywords = {Condensed Matter - Strongly Correlated Electrons, Computer Science - Machine Learning, Physics - Computational Physics},
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}
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@article{hamiltonian-review,
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title = {A critical review of machine learning interatomic potentials and {Hamiltonian}},
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volume = {5},
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issn = {ISSN 2770-372X},
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url = {https://www.oaepublish.com/articles/jmi.2025.17},
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doi = {10.20517/jmi.2025.17},
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abstract = {Machine learning interatomic potentials (ML-IAPs) and machine learning Hamiltonian (ML-Ham) have revolutionized atomistic and electronic structure simulations by offering near ab initio accuracy across extended time and length scales. In this Review, we summarize recent progress in these two fields, with emphasis on algorithmic and architectural innovations, geometric equivariance, data efficiency strategies, model-data co-design, and interpretable AI techniques. In addition, we discuss key challenges, including data fidelity, model generalizability, computational scalability, and explainability. Finally, we outline promising future directions, such as active learning, multi-fidelity frameworks, scalable message-passing architectures, and methods for enhancing interpretability, which is particularly crucial for the field of AI for Science (AI4S). The integration of these advances is expected to accelerate materials discovery and provide deeper mechanistic insights into complex material and physical systems.},
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language = {en},
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number = {4},
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urldate = {2026-05-04},
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journal = {Journal of Materials Informatics},
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author = {Li, Yifan and Zhang, Xiuying and Shen, Lei},
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month = jul,
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year = {2025},
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pages = {N/A--N/A},
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}
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@article{nequip,
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title = {E(3)-equivariant graph neural networks for data-efficient and accurate interatomic potentials},
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volume = {13},
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copyright = {2022 The Author(s)},
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issn = {2041-1723},
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url = {https://www.nature.com/articles/s41467-022-29939-5},
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doi = {10.1038/s41467-022-29939-5},
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abstract = {This work presents Neural Equivariant Interatomic Potentials (NequIP), an E(3)-equivariant neural network approach for learning interatomic potentials from ab-initio calculations for molecular dynamics simulations. While most contemporary symmetry-aware models use invariant convolutions and only act on scalars, NequIP employs E(3)-equivariant convolutions for interactions of geometric tensors, resulting in a more information-rich and faithful representation of atomic environments. The method achieves state-of-the-art accuracy on a challenging and diverse set of molecules and materials while exhibiting remarkable data efficiency. NequIP outperforms existing models with up to three orders of magnitude fewer training data, challenging the widely held belief that deep neural networks require massive training sets. The high data efficiency of the method allows for the construction of accurate potentials using high-order quantum chemical level of theory as reference and enables high-fidelity molecular dynamics simulations over long time scales.},
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language = {en},
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number = {1},
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urldate = {2026-05-04},
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journal = {Nature Communications},
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author = {Batzner, Simon and Musaelian, Albert and Sun, Lixin and Geiger, Mario and Mailoa, Jonathan P. and Kornbluth, Mordechai and Molinari, Nicola and Smidt, Tess E. and Kozinsky, Boris},
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month = may,
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year = {2022},
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keywords = {Atomistic models, Computational chemistry, Computational methods, Computer science, Molecular dynamics},
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pages = {2453},
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}
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@misc{mace,
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title = {{MACE}: {Higher} {Order} {Equivariant} {Message} {Passing} {Neural} {Networks} for {Fast} and {Accurate} {Force} {Fields}},
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shorttitle = {{MACE}},
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url = {http://arxiv.org/abs/2206.07697},
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doi = {10.48550/arXiv.2206.07697},
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abstract = {Creating fast and accurate force fields is a long-standing challenge in computational chemistry and materials science. Recently, several equivariant message passing neural networks (MPNNs) have been shown to outperform models built using other approaches in terms of accuracy. However, most MPNNs suffer from high computational cost and poor scalability. We propose that these limitations arise because MPNNs only pass two-body messages leading to a direct relationship between the number of layers and the expressivity of the network. In this work, we introduce MACE, a new equivariant MPNN model that uses higher body order messages. In particular, we show that using four-body messages reduces the required number of message passing iterations to just two, resulting in a fast and highly parallelizable model, reaching or exceeding state-of-the-art accuracy on the rMD17, 3BPA, and AcAc benchmark tasks. We also demonstrate that using higher order messages leads to an improved steepness of the learning curves.},
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urldate = {2026-05-04},
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publisher = {arXiv},
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author = {Batatia, Ilyes and Kovács, Dávid Péter and Simm, Gregor N. C. and Ortner, Christoph and Csányi, Gábor},
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month = jan,
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year = {2023},
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note = {arXiv:2206.07697},
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keywords = {Statistics - Machine Learning, Condensed Matter - Materials Science, Computer Science - Machine Learning, Physics - Chemical Physics},
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}
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@article{deephe3,
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title = {General framework for {E}(3)-equivariant neural network representation of density functional theory {Hamiltonian}},
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volume = {14},
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issn = {2041-1723},
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url = {https://www.ncbi.nlm.nih.gov/pmc/articles/PMC10199065/},
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doi = {10.1038/s41467-023-38468-8},
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abstract = {The combination of deep learning and ab initio calculation has shown great promise in revolutionizing future scientific research, but how to design neural network models incorporating a priori knowledge and symmetry requirements is a key challenging subject. Here we propose an E(3)-equivariant deep-learning framework to represent density functional theory (DFT) Hamiltonian as a function of material structure, which can naturally preserve the Euclidean symmetry even in the presence of spin–orbit coupling. Our DeepH-E3 method enables efficient electronic structure calculation at ab initio accuracy by learning from DFT data of small-sized structures, making the routine study of large-scale supercells ({\textgreater}104 atoms) feasible. The method can reach sub-meV prediction accuracy at high training efficiency, showing state-of-the-art performance in our experiments. The work is not only of general significance to deep-learning method development but also creates opportunities for materials research, such as building a Moiré-twisted material database., Fundamental symmetries are crucial to the deep-learning modeling of physical systems. Here the authors use equivariant neural networks preserving the Euclidean symmetries to accelerate electronic structure calculations by orders of magnitude keeping sub-meV accuracy.},
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urldate = {2026-05-04},
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journal = {Nature Communications},
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author = {Gong, Xiaoxun and Li, He and Zou, Nianlong and Xu, Runzhang and Duan, Wenhui and Xu, Yong},
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month = may,
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year = {2023},
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pmid = {37208320},
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pmcid = {PMC10199065},
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pages = {2848},
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}
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@inproceedings{qhnet,
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title = {Efficient and {Equivariant} {Graph} {Networks} for {Predicting} {Quantum} {Hamiltonian}},
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url = {https://proceedings.mlr.press/v202/yu23i.html},
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abstract = {We consider the prediction of the Hamiltonian matrix, which finds use in quantum chemistry and condensed matter physics. Efficiency and equivariance are two important, but conflicting factors. In this work, we propose a SE(3)-equivariant network, named QHNet, that achieves efficiency and equivariance. Our key advance lies at the innovative design of QHNet architecture, which not only obeys the underlying symmetries, but also enables the reduction of number of tensor products by 92\%. In addition, QHNet prevents the exponential growth of channel dimension when more atom types are involved. We perform experiments on MD17 datasets, including four molecular systems. Experimental results show that our QHNet can achieve comparable performance to the state of the art methods at a significantly faster speed. Besides, our QHNet consumes 50\% less memory due to its streamlined architecture. Our code is publicly available as part of the AIRS library (https://github.com/divelab/AIRS).},
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language = {en},
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urldate = {2026-05-04},
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booktitle = {Proceedings of the 40th {International} {Conference} on {Machine} {Learning}},
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publisher = {PMLR},
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author = {Yu, Haiyang and Xu, Zhao and Qian, Xiaofeng and Qian, Xiaoning and Ji, Shuiwang},
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month = jul,
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year = {2023},
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pages = {40412--40424},
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}
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@inproceedings{ham-diff,
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title={Neural Hamiltonian Diffusions for Modeling Structured Geometric Dynamics},
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author={Sungwoo Park},
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booktitle={The Thirty-ninth Annual Conference on Neural Information Processing Systems},
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year={2026},
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url={https://openreview.net/forum?id=VswQY0peMr}
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}
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@article{deepmd,
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title = {{DeePMD}-kit: {A} deep learning package for many-body potential energy representation and molecular dynamics},
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volume = {228},
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issn = {00104655},
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shorttitle = {{DeePMD}-kit},
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url = {https://linkinghub.elsevier.com/retrieve/pii/S0010465518300882},
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doi = {10.1016/j.cpc.2018.03.016},
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language = {en},
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urldate = {2026-05-04},
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journal = {Computer Physics Communications},
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author = {Wang, Han and Zhang, Linfeng and Han, Jiequn and E, Weinan},
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month = jul,
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year = {2018},
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pages = {178--184},
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}
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@article{spingnn,
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title = {Spin-dependent graph neural network potential for magnetic materials},
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volume = {109},
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issn = {2469-9950, 2469-9969},
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url = {https://link.aps.org/doi/10.1103/PhysRevB.109.144426},
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doi = {10.1103/PhysRevB.109.144426},
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language = {en},
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number = {14},
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urldate = {2026-05-04},
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journal = {Physical Review B},
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author = {Yu, Hongyu and Zhong, Yang and Hong, Liangliang and Xu, Changsong and Ren, Wei and Gong, Xingao and Xiang, Hongjun},
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month = apr,
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year = {2024},
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pages = {144426},
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}
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@article{ace,
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title = {Atomic cluster expansion for accurate and transferable interatomic potentials},
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volume = {99},
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issn = {2469-9950, 2469-9969},
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url = {https://link.aps.org/doi/10.1103/PhysRevB.99.014104},
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doi = {10.1103/PhysRevB.99.014104},
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language = {en},
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number = {1},
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urldate = {2026-05-04},
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journal = {Physical Review B},
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author = {Drautz, Ralf},
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month = jan,
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year = {2019},
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pages = {014104},
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}
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@misc{ai2dft,
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title = {Neural-network {Density} {Functional} {Theory} {Based} on {Variational} {Energy} {Minimization}},
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url = {http://arxiv.org/abs/2403.11287},
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doi = {10.48550/arXiv.2403.11287},
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abstract = {Deep-learning density functional theory (DFT) shows great promise to significantly accelerate material discovery and potentially revolutionize materials research. However, current research in this field primarily relies on data-driven supervised learning, making the developments of neural networks and DFT isolated from each other. In this work, we present a theoretical framework of neural-network DFT, which unifies the optimization of neural networks with the variational computation of DFT, enabling physics-informed unsupervised learning. Moreover, we develop a differential DFT code incorporated with deep-learning DFT Hamiltonian, and introduce algorithms of automatic differentiation and backpropagation into DFT, demonstrating the capability of neural-network DFT. The physics-informed neural-network architecture not only surpasses conventional approaches in accuracy and efficiency, but also offers a new paradigm for developing deep-learning DFT methods.},
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urldate = {2026-05-04},
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publisher = {arXiv},
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author = {Li, Yang and Tang, Zechen and Chen, Zezhou and Sun, Minghui and Zhao, Boheng and Li, He and Tao, Honggeng and Yuan, Zilong and Duan, Wenhui and Xu, Yong},
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month = aug,
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year = {2024},
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note = {arXiv:2403.11287},
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keywords = {Physics - Computational Physics, Condensed Matter - Materials Science},
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}
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