Xiaofeng Xu

I am currently a Postdoctoral Fellow in Applied Mathematics and Computational Science (AMCS) in the Department of Mathematics at the University of Pennsylvania, working with Prof. Shanyin Tong. 

I was a postdoctoral researcher at Pennsylvania State University, working with Prof. Wenrui Hao from Oct. 2025 to Jul. 2026.

I obtained my Ph.D. in Mathematics from King Abdullah University of Science and Technology (KAUST), where I was advised by Prof. Jinchao Xu.

My research focuses on developing efficient and provably convergent training algorithms for neural network optimization in solving partial differential equations (PDEs), bridging theoretical analysis and modern machine learning. I have also worked on numerical methods for phase-field simulations and simulating complex physical phenomena in materials science.

Education:

• Ph.D., Applied Mathematics and Computational Sciences, King Abdullah University of Science and Technology, 2022 - 2025

Advisor: Prof. Jinchao Xu, Research on neural network optimizaiton for solving PDEs

Thesis: Accurate and efficient training algorithms for neural network optimization in numerical PDEs

• M.S., Mathematics, Pennsylvania State University, 2019-2022

Advisor: Prof. Jinchao Xu, Research on phase-field simulations

• B.S., Mathematics and Computer  Science, Hong Kong University of Science and Technology, 2015 -2019

Publications:

1. J. Xu, X. Xu, Randomized Greedy Algorithms for Neural Network Optimization in Solving Partial Differential Equations, Journal of Scientific Computing, 2025

2. Y. Shi, X. Xu, X. Cheng, S. Shenoy, J. Xu, L.Q. Chen, Q-POP-IMT: An Open-Source Phase-Field Software for Simulating Insulator–Metal Transition Processes in Quantum Materials, Computer Physics Communications, 2025

3. W. Hao, S. Lee, X. Xu, Z. Xu, Stability and Robustness of Time-Discretization Schemes for the Allen–Cahn Equation via Bifurcation and Perturbation Analysis, Journal of Computational Physics, 2024

4. J. Park, J. Xu, X. Xu, A Neuron-Wise Subspace Correction Method for the Finite Neuron Method, Journal of Computational Mathematics, 2024

5. J. Xu, X. Xu, Lack of Robustness and Accuracy for Many Numerical Schemes in Phase-Field Simulations, Mathematical Models and Methods in Applied Sciences (M3AS), 2023

6. X. Xu, L. Zhang, Y. Shi, L.Q. Chen, J. Xu, Integral Boundary Conditions in Phase-Field Models, Computers & Mathematics with Applications, 2023

Preprint:

1. T. Mao, J. Xu, X. Xu, Solving High-Dimensional PDEs Using Linearized Neural Networks, arXiv preprint arXiv:2601.11771 (2026)

2. X. Xu, T. Dan, Z. Zhou, B. Li, G. Wu, W. Hao, Neural operator-based digital twins for modeling amyloid-$\beta$ and tau propagation and treatment optimization in Alzheimer's disease. arXiv preprint arXiv:2606.25185. (2026).

Teaching:

• I will be teaching AMCS 6025 Numerical and Applied Analysis, University of Pennsylvania, Fall 2026.  

• Instructor, Math 451 Numerical Computations, Penn State University, Spring 2026

• Teaching Assistant, Multigrid Methods, KAUST, Spring 2025

• Teaching Assistant, Mathematical Economics and Finance, KAUST, Spring 2024

• Teaching Assistant, Scientific Machine Learning Summer School, CIRM, France, Summer 2023

• Instructor, Math 250 Ordinary Differential Equations, Penn State University, Fall 2021

Presentation & Posters:

Talks: 

• CRUNCH Seminar at Brown University, Feb 2026. Link: https://www.youtube.com/watch?v=Z-zQK2KRar0 

• Randomized Greedy Algorithms for Neural Network Optimization, IMG Conference, KAUST, Saudi Arabia, Feb 2025 (30-minute talk)

• Randomized Greedy Algorithms for Neural Network Optimization, SciCADE Conference, NUS, Singapore, Jul 2024 (30-minute talk)

• Integral Boundary Conditions in Phase-Field Models, FEniCS Conference, Jun 2024

• A Neuron-Wise Subspace Correction Method for the Finite Neuron Method, CBMS Conference on Deep Learning and Numerical PDEs, Morgan State University, Jun 2023

• Integral Boundary Conditions in Phase-Field Models, Finite Element Circus, University of Florida, Apr 2022

• Lack of Robustness and Accuracy for Many Numerical Schemes in Phase-Field Simulations, Finite Element Circus, Penn State University, Nov 2021

• Deep Learning for Dermatology, Joint Mathematics Meeting, San Diego, Jan 2018

Posters:

• Random Discretization of Dictionaries: A Practical Approach for Greedy Algorithms for Neural Network Approximations, Stochastic Numerics and Statistical Learning Workshop, KAUST, May 2024

• Neuron-Wise Subspace Correction Method for ReLU Neural Networks for PDEs, SCML Conference Poster Session, KAUST, Nov 2022