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  • Open Access

Probabilistic Denoising for Reliable Signal Extraction in Spectroscopy

Younsik Kim* and Changyoung Kim†

  • *Contact author: leblang@snu.ac.kr
  • †Contact author: changyoung@snu.ac.kr
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PRX Intelligence 1, 023005 – Published 20 October, 2026

DOI: https://doi.org/10.1103/yh1d-64cc

Abstract

While deep learning offers powerful capabilities for scientific research, its application is often hindered by a lack of quantitative reliability. To address this, we introduce a probabilistic denoising framework that simultaneously extracts denoised signals and element-wise predictive uncertainties from noisy data. We demonstrate this approach on three-dimensional angle-resolved photoemission spectroscopy data, showing that the model reliably recovers the spectral features of a cuprate superconductor from Poisson-distributed noise with an average count of only 0.02 electrons per voxel. Crucially, we show that these predicted uncertainties can be propagated into subsequent superconducting gap analyses, enabling quantitative parameter extraction with scientifically meaningful error bars. Furthermore, we validate the broad applicability of our approach by successfully extending it to two-dimensional x-ray diffraction data. Ultimately, this approach establishes uncertainty-aware deep learning not merely as a visualization tool, but as a rigorous framework for scientific data analysis.

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References (39)

  1. R. Roscher, B. Bohn, M. F. Duarte, and J. Garcke, Explainable machine learning for scientific insights and discoveries, IEEE Access 8, 42200 (2020).
  2. M. Abdar, F. Pourpanah, S. Hussain, D. Rezazadegan, L. Liu, M. Ghavamzadeh, P. Fieguth, X. Cao, A. Khosravi, U. R. Acharya, et al., A review of uncertainty quantification in deep learning: Techniques, applications and challenges, Inf. Fusion 76, 243 (2021).
  3. G. E. Karniadakis, I. G. Kevrekidis, L. Lu, P. Perdikaris, S. Wang, and L. Yang, Physics-informed machine learning, Nat. Rev. Phys. 3, 422 (2021).
  4. A. Tarantola, Inverse Problem Theory and Methods for Model Parameter Estimation (SIAM, Philadelphia, PA, 2005).
  5. M. Bertero, T. A. Poggio, and V. Torre, Ill-posed problems in early vision, Proc. IEEE 76, 869 (1988).
  6. A. Kendall and Y. Gal, What uncertainties do we need in Bayesian deep learning for computer vision? Adv. Neural Inf. Process. Syst. 30, 5574 (2017).
  7. J. Gawlikowski, C. R. N. Tassi, M. Ali, J. Lee, M. Humt, J. Feng, A. Kruspe, R. Triebel, P. Jung, R. Roscher, et al., A survey of uncertainty in deep neural networks, Artif. Intell. Rev. 56, 1513 (2023).
  8. A. F. Psaros, X. Meng, Z. Zou, L. Guo, and G. E. Karniadakis, Uncertainty quantification in scientific machine learning: Methods, metrics, and comparisons, J. Comput. Phys. 477, 111902 (2023).
  9. A. Ghosh, B. G. Sumpter, O. Dyck, S. V. Kalinin, and M. Ziatdinov, Ensemble learning-iterative training machine learning for uncertainty quantification and automated experiment in atom-resolved microscopy, npj Comput. Mater. 7, 100 (2021).
  10. R. K. Vasudevan, M. Ziatdinov, L. Vlcek, and S. V. Kalinin, Off-the-shelf deep learning is not enough, and requires parsimony, Bayesianity, and causality, npj Comput. Mater. 7, 16 (2021).
  11. J. Masci, U. Meier, D. Cireşan, and J. Schmidhuber, Stacked convolutional auto-encoders for hierarchical feature extraction, in International Conference on Artificial Neural Networks (Springer, Berlin, Heidelberg, 2011), pp. 52–59.
  12. See Supplemental Material at https://link.aps.org/supplemental/10.1103/yh1d-64cc for details of the network architecture and training, momentum distribution curve analysis, statistical calibration of the predicted uncertainties, and an additional demonstration of denoising on Bi2Te3 thin-film data.
  13. Y. Kim, D. Oh, S. Huh, D. Song, S. Jeong, J. Kwon, M. Kim, D. Kim, H. Ryu, J. Jung, et al., Deep learning-based statistical noise reduction for multidimensional spectral data, Rev. Sci. Instrum. 92, 073901 (2021).
  14. M. Seitzer, A. Tavakoli, D. Antic, and G. Martius, On the pitfalls of heteroscedastic uncertainty estimation with probabilistic neural networks, International Conference on Learning Representations, arXiv:2203.09168.
  15. S. Ý. Ágústsson, M. Bianchi, and H. Philip, 3D ARPES angle scan collection, Zenodo, 2024, http://doi.org/10.5281/zenodo.12665275.
  16. W. H. Press, Numerical Recipes: The Art of Scientific Computing (Cambridge University Press, Cambridge, UK, 2007), 3rd ed.
  17. M. R. Norman, M. Randeria, H. Ding, and J. Campuzano, Phenomenology of the low-energy spectral function in high-Tc superconductors, Phys. Rev. B 57, R11093 (1998).
  18. D. Wu, J. Jia, J. Yang, W. Hong, Y. Shu, T. Miao, H. Yan, H. Rong, P. Ai, X. Zhang, et al., Nodal s±pairing symmetry in an iron-based superconductor with only hole pockets, Nat. Phys. 20, 571 (2024).
  19. B. D. Cullity and R. Smoluchowski, Elements of X-ray diffraction, Phys. Today 10(3), 50 (1957).
  20. J. Oppliger, M. M. Denner, J. Küspert, R. Frison, Q. Wang, A. Morawietz, O. Ivashko, A.-C. Dippel, M. v. Zimmermann, I. Biało, et al., Weak signal extraction enabled by deep neural network denoising of diffraction data, Nat. Mach. Intell. 6, 180 (2024).
  21. J. Oppliger, M. M. Denner, J. Küspert, R. Frison, Q. Wang, A. Morawietz, O. Ivashko, A.-C. Dippel, M. v. Zimmermann, N. B. Christensen, T. Kurosawa, N. Momono, M. Oda, F. D. Natterer, M. H. Fischer, T. Neupert, and J. Chang, X-ray diffraction dataset for experimental noise filtering, Zenodo, 2022, http://doi.org/10.5281/zenodo.10816877.
  22. J. Oppliger, X-ray diffraction denoising using deep convolutional neural networks, Zenodo, 2023, http://doi.org/10.5281/zenodo.10245374.
  23. S. M. Kay, Fundamentals of Statistical Signal Processing: Estimation Theory (Prentice Hall, Englewood Cliffs, NJ, 1993), Vol. 1.
  24. T. M. Cover, Elements of Information Theory (John Wiley & Sons, New York, NY, 1999).
  25. H. Peng, X. Gao, Y. He, Y. Li, Y. Ji, C. Liu, S. A. Ekahana, D. Pei, Z. Liu, Z. Shen, et al., Super resolution convolutional neural network for feature extraction in spectroscopic data, Rev. Sci. Instrum. 91, 033905 (2020).
  26. T. C. Meyer, G.-R. Siemann, P. Majchrzak, T. Seyller, J. Rigden, Y. Zhang, E. Springate, C. Sanders, and P. Hofmann, Line shapes in time-and angle-resolved photoemission spectroscopy explored by machine learning, Electron. Struct. 7, 045001 (2025).
  27. S. Ý. Ágústsson, M. A. Haque, T. T. Truong, M. Bianchi, N. Klyuchnikov, D. Mottin, P. Karras, and P. Hofmann, An autoencoder for compressing angle-resolved photoemission spectroscopy data, Mach. Learn.: Sci. Technol. 6, 015019 (2025).
  28. P. Majchrzak, C. Sanders, Y. Zhang, A. Kuibarov, O. Suvorov, E. Springate, I. Kovalchuk, S. Aswartham, G. Shipunov, B. Büchner, et al., Machine-learning approach to understanding ultrafast carrier dynamics in the three-dimensional Brillouin zone of PtBi2, Phys. Rev. Res. 7, 013025 (2025).
  29. X. Chen, Y. Sun, E. Hruska, V. Dixit, J. Yang, Y. He, Y. Wang, and F. Liu, Detecting thermodynamic phase transition via explainable machine learning of photoemission spectroscopy, Newton 1, 100066 (2025).
  30. T. Yoon, S. W. Kim, H. Byun, Y. Kim, C. D. Carter, and H. Do, Deep learning-based denoising for fast time-resolved flame emission spectroscopy in high-pressure combustion environment, Combust. Flame 248, 112583 (2023).
  31. F. Joucken, J. L. Davenport, Z. Ge, E. A. Quezada-Lopez, T. Taniguchi, K. Watanabe, J. Velasco Jr, J. Lagoute, and R. A. Kaindl, Denoising scanning tunneling microscopy images of graphene with supervised machine learning, Phys. Rev. Mater. 6, 123802 (2022).
  32. R. P. Xian, V. Stimper, M. Zacharias, M. Dendzik, S. Dong, S. Beaulieu, B. Schölkopf, M. Wolf, L. Rettig, C. Carbogno, et al., A machine learning route between band mapping and band structure, Nat. Comput. Sci. 3, 101 (2023).
  33. J. Liu, D. Huang, Y.-F. Yang, and T. Qian, Removing grid structure in angle-resolved photoemission spectra via deep learning method, Phys. Rev. B 107, 165106 (2023).
  34. S. A. Ekahana, G. I. Winata, Y. Soh, A. Tamai, R. Milan, G. Aeppli, and M. Shi, Transfer learning application of self-supervised learning in ARPES, Mach. Learn.: Sci. Technol. 4, 035021 (2023).
  35. M. Na, C. Zhou, S. K. Dufresne, M. Michiardi, and A. Damascelli, A simulation-based training framework for machine-learning applications in ARPES, arXiv:2508.15983.
  36. M. Ziatdinov, A. Ghosh, C. Y. Wong, and S. V. Kalinin, Atomai framework for deep learning analysis of image and spectroscopy data in electron and scanning probe microscopy, Nat. Mach. Intell. 4, 1101 (2022).
  37. I. Loshchilov and F. Hutter, Decoupled weight decay regularization, International Conference on Learning Representations, arXiv:1711.05101.
  38. Y. Kim, Bi-2212 test data for probabilistic denoising for reliable signal extraction in spectroscopy, Figshare, 2026, doi: 10.6084/m9.figshare.31941057.
  39. Y. Kim, Code for training neural network in probabilistic denoising for reliable signal extraction in spectroscopy, Figshare, 2026, doi: 10.6084/m9.figshare.31939203.

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PRX Intelligence 1, 023005– Published 20 October, 2026
  • Received 7 April 2026
  • Accepted 31 August 2026
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Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article's title, journal citation, and DOI.

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