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Photoelectron spectroscopy serves as a prevalent characterization technique in the field of materials science. Especially, angle-resolved photoelectron spectroscopy (ARPES) provides a direct method for determining the energy-momentum dispersion relationship and Fermi surface structure of electrons in a material system, therefore ARPES has become a potent tool for investigating many-body interactions and correlated quantum materials. With the emergence of technologies such as time-resolved ARPES and nano-ARPES, the field of photoelectron spectroscopy continues to advance. Meanwhile, the development of synchrotron radiation facilities has led to an increase of high-throughput and high-dimensional experimental data. This highlights the urgency for developing more efficient and accurate data processing methods, as well as extracting deeper physical information. In light of these developments, machine learning will play an increasingly significant role in various fields, including but not limited to ARPES. This paper reviews the applications of machine learning in photoelectron spectroscopy, mainly including the following three aspects. 1) Data Denoising Machine learning can be utilized for denoising photoelectron spectroscopy data. The denoising process via machine learning algorithms can be divided into two methods. Neither of the two methods need manual data annotation. The first method is to use noise generation algorithms to simulate experimental noise, so as to obtain effective low signal-to-noise ratio data pair to high signal-to-noise ratio data pair. And the second method is to extract noise and clean spectral data. 2) Electronic Structure and Chemical Composition Analysis Machine learning can be used for analyzing electronic structure and chemical composition. (Angle-resolved) photoelectron spectroscopy contains abundant information about material structure. Information such as energy band structure, self-energy, binding energy, and other condensed matter data can be rapidly acquired through machine learning schemes. 3) Prediction of Photoelectron Spectroscopy The electronic structure information obtained by combining first-principles calculation can also predict the photoelectron spectroscopy. The rapid acquisition of photoelectron spectroscopy data through machine learning algorithms also holds significance for material design. Photoelectron spectroscopy holds significant importance in the study of condensed matter physics. In the context of the development of synchrotron radiation, the construction of an automated data acquisition and analysis system can play a pivotal role in studying condensed matter physics. In addition, adding more physical constraints to the machine learning model will improve the interpretability and accuracy of the model. There exists a close relationship between photoelectron spectroscopy and first-principles calculations of electronic structure properties. The integration of these two through machine learning is anticipated to significantly contribute to the study of electronic structure properties. Furthermore, as machine learning algorithms continue to evolve, the application of more advanced machine learning algorithms in photoelectron spectroscopy research is expected. Building automated data acquisition and analysis systems, designing comprehensive workflows based on machine learning and first-principles methods, and integrating new machine learning techniques will help accelerate the progress of photoelectron spectroscopy experiments and facilitate the analysis of electronic structure properties and microscopic physical mechanisms, thereby advancing the frontier research in quantum materials and condensed matter physics. 
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图 1 (a)近年来光电子能谱相关文章的发文数量; (b)光电子能谱常见应用领域(来源: web of science——以光电子能谱相关关键词搜索获得的结果); (c)光电子能谱常见的研究体系
Figure 1. (a) The number of papers related to photoelectron spectroscopy in recent years; (b) common application fields of photoelectron spectroscopy (Source: web of science—Results obtained by searching for keywords related to photoelectron spectroscopy); (c) common research systems of photoelectron spectroscopy.
图 2 (a)光电子激发和采集示意图; (b)光电子强度和电子态密度的关系; (c)材料内部光电子平均自由程与光子能量的关系. (b)引用自参考文献[14], 版权属于 Springer Nature; (c)引用自参考文献[35], 版权属于 John Wiley and Sons
Figure 2. (a) Photoelectron excitation and collection schematics; (b) relationship between photoelectron intensity and electron density of states; (c) relationship between the average free path of photoelectrons inside the material and photon energy. Panel (b) reprinted with permission from Ref. [14], copyright 2019 by the Springer Nature; panel (c) reprinted with permission from Ref. [35], copyright 1979 by the John Wiley and Sons.
图 3 机器学习在光电子能谱中的作用. 机器学习的应用主要分为四个方面, 分别是对光电子能谱数据进行降噪; 加速元素分析; 提取光电子能谱中的物理信息(如电子结构信息); 以及通过结合理论计算的结果预测材料的光电子能谱
Figure 3. Role of machine learning in photoelectron spectroscopy. The application of machine learning is mainly divided into four aspects: noise reduction of photoelectron spectroscopy data; accelerated elemental analysis; extraction of physical information in the photoelectron spectroscopy (such as the electronic structure information); and the photoelectron spectroscopy of the material is predicted by combining the results of theoretical calculations.
图 4 光电子能谱数据降噪中的机器学习方法. 方法一: 生成噪声数据模拟实验噪声, 从而进行降噪网络的训练[60,61]; 方法二: 通过不同的网络分别提取噪声和干净的光谱数据, 然后将两者组合形成生成数据. 因此, 两种方法的损失函数都是通过评估生成数据与原始数据的相似性[57,62]
Figure 4. Machine learning methods in noise reduction of photoelectron spectroscopy data. Method 1: noise data is generated to simulate the noise, so as to train the noise reduction network[60,61]; method 2: noise and clean spectral data are extracted by different networks, and then combined to form the generated data. Therefore, the loss function of both methods is to evaluate the similarity between the generated data and the original data[57,62].
图 5 聚类数为8时k-means的结果 (a)不同簇数时, 每个簇的空间分布; (b)对每个簇中的簇成员进行平均得到的平均EDC. 引用自参考文献[64], 版权属于 Springer Nature
Figure 5. Results of k-means when the number of clusters is 8: (a) Spatial distribution of each of clusters for different number of clusters; (b) mean-EDCs obtained by averaging the cluster members in each cluster. Reprinted with permission from Ref. [64], copyright 2022 by the Springer Nature.
图 6 (a)马尔可夫随机场进行能带重建过程: 实验获得的ARPES数据经过预处理和第一性原理计算的初始值输入到马尔可夫随机场中, 得到的结果经过后处理便能形成按能带指数排列的光电发射色散面, 即能带结构; (b)重建的14层价带; (c)重建出的能带色散(红色线条)与在光电子能带映射数据的叠加. 引用自参考文献[69], 版权属于 Springer Nature
Figure 6. (a) Band reconstruction process with Markov random field: The ARPES data obtained from the experiment are preprocessed, and the initial values of the first-principles calculation are input into the Markov random field. The obtained results are post-processed to form a photoelectric emission dispersion surface arranged exponentially according to the energy band, that is, the band structure; (b) the reconstructed 14-layer valence band; (c) the superposition of the reconstructed band dispersion (red line) and the data mapped in the photoelectron energy band. Reprinted with permission from Ref. [69], copyright 2022 by the Springer Nature.
图 7 自能提取的机器学习的流程, 用于从实验观测的光电子谱函数$ A({\boldsymbol{k}}, \omega) $中提取正常自能和反常自能. 引用自参考文献[73], 版权属于美国物理学会
Figure 7. Flow chart of machine-learning procedure. It is used to extract normal self-energy $ {{\varSigma}}({\boldsymbol{k}}, \omega)^\text{nor} $ and anomalous self-energy $ {{\varSigma}}({\boldsymbol{k}}, \omega)^\text{ano} $ from the experimentally observed spectral function $ A({\boldsymbol{k}}, \omega) $. Reprinted with permission from Ref. [73], copyright 2021 by the American Physical Society.
图 8 使用原子位置平滑重叠和核岭回归预测结合能和XPS (a)使用SOAP多体描述符处理CHO材料数据库获得的基于聚类的多维缩放图; (b)—(d) a-COx的C 1s谱. 其中浅灰色的C原子贡献了光谱中的浅灰色区域, 而深灰色的C原子贡献了光谱中的深灰色区域. 引用自参考文献[97], 版权属于美国化学会
Figure 8. Smooth overlap of atomic positions and kernel ridge regression are used to predict the binding energy and XPS: (a) Using SOAP multi-body descriptor to process the cluster-based multidimensional scaling map obtained from CHO material database; (b)–(d) the C 1s spectra of a-COx, the light gray C atoms contribute to the light gray region in the spectrum, while the dark gray C atoms contribute to the dark gray region in the spectrum. Reprinted with permission from Ref. [97], copyright 2022 by the American Chemical Society.
图 9 (a) $ {\rm{InAs}}(001) \ \beta 2(2 \times 4) $ 俯视图和侧视图; (b), (c), (d) 分别为 $ {\rm{InAs}}(001) \ \beta 2(2 \times 4) $能带结构的xy-unfolding, z-unfolding, bulk unfolding; (e) $ {\rm{InAs}}(001) \ \beta 2(2 \times 4) $ 表面的布里渊区. 引用自参考文献[99], 版权属于 John Wiley and Sons
Figure 9. (a) $ {\rm{InAs}}(001) \ \beta 2(2 \times 4) $ top view and side view; (b), (c), (d) xy-unfolding, z-unfolding, bulk unfolding of $ {\rm{InAs}}(001) \ \beta 2 $$ (2 \times 4) $ band structure, respectively; (e) the Brillouin zone of $ {\rm{InAs}}(001) \ \beta 2(2 \times 4) $ surface. Reprinted with permission from Ref. [99], copyright 2022 by the John Wiley and Sons.
图 10 现有的以及未来可用于角分辨光电子能谱的机器学习方法
Figure 10. Existing and future machine learning methods for angle-resolved photoelectron spectroscopy.
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