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密度泛函理论（Density Functional Theory, DFT）在当代电子结构计算中占据主导地位，然而其计算复杂度随体系规模呈立方增长，制约了其在复杂体系或高精度计算中的应用。近年来，机器学习（Machine Learning，ML）与第一性原理计算的结合，为这一问题提供了新的解决方案。本文对ML加速电子结构计算的方法进行了综述,重点讨论现有研究在加速材料电子结构计算中所取得的重要进展。此外,对未来研究中基于ML技术进一步克服电子结构计算的精度和效率瓶颈、扩展适用范围、实现在大尺度材料体系中计算模拟与实验测量的深度融合做了展望。Density functional theory (DFT) stands as the predominant workhorse for electronic structure calculation across physics, chemistry, and materials science. However, its practical application is fundamentally constrained by a computational cost that scales cubically with system size, rendering high-precision studies of complex or large-scale materials prohibitively expensive. This review addresses the pivotal challenge by surveying the rapidly evolving paradigm of integrating machine learning (ML) with first-principles calculations to dramatically accelerate and scale electronic structure prediction. Our primary objective is to provide a comprehensive and critical overview of the methodological advances, physical outcomes, and transformative potential of this interdisciplinary field.
The core methodological progression involves a shift from black-box property predictors to symmetry-preserving, transferable models that learn the fundamental Hamiltonian—the central quantity from which diverse electronic properties derive. We detail this evolution, beginning with pioneering applications in molecular systems using graph neural networks (e.g., SchNOrb, DimeNet) to predict energies, wavefunctions, and Hamiltonian matrices with meV-level accuracy. The review then focuses on the critical extension to periodic solids, where preserving symmetries like E(3)-equivariance and handling vast configurational spaces are paramount. We systematically analyze three leading model families that define the state-of-the-art: the DeepH series, which employs local coordinate message passing and E(3)-equivariant networks to achieve sub-meV accuracy and linear scaling; the HamGNN framework, built on rigorous equivariant tensor decomposition, excelling in modeling systems with spin-orbit coupling and charged defects; and the DeePTB approach, which leverages deep learning for tight-binding Hamiltonian parameterization, enabling quantum-accurate simulations of millions of atoms.
These methods yield significant physical results and computational breakthroughs. Key outcomes include: 1) Unprecedented accuracy and speed. Models consistently achieve Hamiltonian prediction mean absolute errors (MAE) below 1 meV (e.g., DeepH-E3: ~0.4 meV in graphene; HamGNN: ~1.5 meV in QM9 molecules), coupled with computational speedups of 3 to 5 orders of magnitude compared to conventional DFT. 2) Scale bridging. Successful applications now span from small molecules to defect-containing supercells with over 10,000 atoms (e.g., HamGNN-Q on a 13,824-atom GaAs defect) and even to millions of atoms for optoelectronic property simulations (DeePTB). 3) Expanded application scope. The review highlights how these ML-accelerated tools are revolutionizing research in previously intractable areas: predicting spectroscopic properties of molecules (e.g., DetaNet for NMR/UV-Vis spectra), elucidating electronic structures of topological materials and magnetic moiré systems, computing electron-phonon coupling and carrier mobility with DFT-level accuracy but far greater efficiency (HamEPC framework), and enabling high-throughput screening for materials design.
In conclusion, ML-accelerated electronic structure calculation has matured into a powerful paradigm, transitioning from a proof-of-concept to a tool capable of delivering DFT-fidelity results at dramatically reduced cost for systems of realistic scale and complexity. However, challenges remain, including model interpretability ("black-box" nature), transferability to unseen elements, and seamless integration with existing plane-wave DFT databases. Future directions point towards physics-constrained unsupervised learning (e.g., DeepH-zero), development of more universal and element-agnostic architectures, and the creation of closed-loop, artificial intelligence (AI)-driven discovery pipelines. By overcoming current limitations, these methods hold the potential to fundamentally reshape the materials research landscape, accelerating the journey from atomistic simulation to rational material design and discovery.-
Keywords:
- Machine Learning /
- Graph Neural Networks /
- First-Principles /
- Electronic Structure
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