Research

Research Interests

1. Machine Learning

• Scientific machine learning (SciML): physics-informed machine learning, optimization-informed neural networks, and operator learning

•  Mathematics of machine learning: approximation and generalization theory in deep learning, convergence and error estimate for SciML

• Federated learning

• Control and machine learning

2. Scientific Computing

• Optimal control of PDEs and control of dynamical systems

• Inverse problems

• Shape design

• Physics simulations

3. Optimization

• Stochastic optimization

• Operator splitting methods

Honors and Awards:

• Best Thesis Award, The Hong Kong Mathematical Society, 2023.

• Humboldt Research Fellowship for Postdocs, Alexander von Humboldt Foundation, Germany, 2022.

• EASIAM Student Paper Prize (first prize), 2022.

• The Best Paper Prize in the 5th Graduate Forum of the Mathematical Programming Branch of Operational Research Society of China, 2019. 

Publications 

(My co-authored works always list the authors alphabetically to signify equal contributions, except when collaborating with students from the mainland of China, due to their graduation requirements)

• Scientific Computing

• H. Wang, C. Yu, Y. Song “An efficient augmented Lagrangian method for state constrained optimal control problems”, Journal of Optimization Theory and Applications, (2024), pp.1–31.

• Y. Song, X. Yuan, and H. Yue. “A numerical approach to the optimal control of thermally convective flows”, Journal of Computational Physics, 494 (2023), 112458.

• U. Biccari, Y. Song, X. Yuan, and E. Zuazua. “A two-stage numerical approach for the sparse initial source identification of a diffusion-advection equation”, Inverse Problems, 39 (2023), 095003. 

• M. Li, Y. Song, X. Yang, and K. Zhang. ”Lattice structure design optimization under localized linear buckling constraints”, Computers & Structures, 286 (2023), 107112. 

• R. Glowinski, Y. Song, X. Yuan, and H. Yue. “Bilinear optimal control of an advection-reaction-diffusion system”, SIAM Review, 64 (2022), pp. 392–421.

• R. Glowinski, Y. Song, X. Yuan, and H. Yue. ”Application of the alternating direction method of multipliers to control constrained parabolic optimal control problems and beyond.” Annals of Applied Mathematics, 38 (2022), pp. 115–158. 

• Y. Gao, J. Li, Y. Song, C. Wang, and K. Zhang. “Alternating direction-based method for optimal control problem constrained by Stokes equation”, Journal of Inverse and Ill-posed Problems, 30 (2022), pp. 81–99.

• R. Glowinski, Y. Song, and X. Yuan. “An ADMM numerical approach to linear parabolic state constrained optimal control problems”, Numerische Mathematik, 144 (2020), pp. 931–966. 

• Y. Song, X. Yuan, and H. Yue. “An inexact Uzawa algorithmic framework for nonlinear saddle point problems with applications to elliptic optimal control problem”, SIAM Journal on Numerical Analysis, 57 (2019), pp. 2656–2684. 

• K. Zhang,  J.  Li , Y. Song, and X. Wang. “An alternating direction method of multipliers for elliptic equation constrained optimization problem”, Science China Mathematics, 60 (2017), pp. 1–18. 

• Optimization

• Y. Song, Z. Wang, X. Yuan, and H. Yue. ”A single-loop stochastic proximal quasi-Newton method for large-scale nonsmooth convex optimization”, arXiv preprint arXiv:2409.16971.

• Scientific Machine Learning 

  • Y. Gao, Y. Song, Z. Tan, H. Yue, S. Zeng. "Prox-PINNs: A deep learning algorithmic framework for elliptic variational inequalities". arXiv preprint arXiv:2505.14430.

  • Y. Song, X. Yuan, H. Yue, and T. Zeng. ”An operator learning approach to nonsmooth optimal control of nonlinear PDEs", arXiv preprint arXiv:2409.14417.

  • M. Lai, Y. Song, X. Yuan, H. Yue, and T. Zeng. “The hard-constraint PINNs for interface optimal control problems”, SIAM Journal on Scientific Computing, 47 (2025), pp. C601-C629. 

  • Y. Song, X. Yuan, and H. Yue. “The ADMM-PINNs algorithmic framework for nonsmooth PDE-constrained optimization: a deep learning approach”,     SIAM Journal on Scientific Computing, 46 (2024), pp. C659-C687.

  • Y. Song, X. Yuan, and H. Yue. “Accelerated primal-dual methods with enlarged step sizes and operator learning for nonsmooth optimal control problems”, arXiv preprint arXiv:2307.00296.

• Machine Learning

• Y. Song, Z. Wang, and E. Zuazua. “Approximate and weighted data reconstruction attack in federated learning”, arXiv preprint arXiv:2308.06822.

• Y. Song, Z. Wang, and E. Zuazua. “FedADMM-InSa: An inexact and self-adaptive ADMM for federated learning”, Neural Networks, 181 (2025), 106772.