Greetings! I'm a mathematician at Sandia National Labs in New Mexico, in the Center for Computing Research. My research interests include:
- Approximation theory and training algorithms for deep neural networks
- Theory and data-driven modeling for fractional-order and nonlocal partial differential equations
- Scientific machine learning, including physics-informed neural networks and constrained Gaussian processes
- Stochastic processes and Monte Carlo methods for partial differential equations
Here are my publications. If you would like to discuss any of them, feel free to email me.
- Partition of unity networks: deep hp-approximation, with Kookjin Lee, Nathaniel A. Trask, Ravi G. Patel, and Eric C. Cyr. Accepted to AAAI-MLPS 2021. (Preprint)
- Analysis of Anisotropic Nonlocal Diffusion Models: Well-posedness of Fractional Problems for Anomalous Transport, with Marta D'Elia. arXiv Preprint. (Preprint)
- Gaussian Processes Constrained by Boundary Value Problems, with Ari Frankel and Laura Swiler. arXiv Preprint. (Preprint)
- A block coordinate descent optimizer for classification problems exploiting convexity, with Ravi G. Patel, Nathaniel A. Trask, and Eric C. Cyr. Accepted to AAAI-MLPS 2021. (Preprint)
- A Survey of Constrained Gaussian Process Regression: Approaches and Implementation Challenges, with Laura Swiler, Ari Frankel, Cosmin Safta, and John Jakeman. Journal of Machine Learning for Modeling and Computing (2020). (Journal|Preprint)
- Data-driven learning of robust nonlocal physics from high-fidelity synthetic data, with Huaiqian You, Yue Yu, Nathaniel Trask, and Marta D'Elia. Computer Methods in Applied Mechanics and Engineering (2020). (Journal|Preprint)
- A Unified Theory of Fractional, Nonlocal, and Weighted Nonlocal Vector Calculus, with Marta D'Elia, Hayley Olson, and George Em Karniadakis. arXiv Preprint. (Preprint)
- Robust training and initialization of deep neural networks: an adaptive basis viewpoint, with Eric C. Cyr, Ravi G. Patel, Mauro Perego, and Nathaniel A. Trask. Mathematical and Scientific Machine Learning (2020). (Journal|Preprint)
- Stochastic solution of elliptic and parabolic boundary value problems for the spectral fractional Laplacian, with Guofei Pang. arXiv Preprint. (Preprint)
- Machine learning of space-fractional differential equations, with Maziar Raissi, Paris Perdikaris, and George Em Karniadakis. SIAM Journal on Scientific Computing (2019). (Journal|Preprint)
What is the Fractional Laplacian? A comparative review with new results,
with Anna Lischke, Guofei Pang, Fangying Song, Christian Glusa, Xiaoning Zheng, Zhiping Mao, Wei Cai,
Mark M. Meerschaert, Mark Ainsworth, and George Em Karniadakis.
Journal of Computational Physics (2019).
- Fractional path integral Monte Carlo, with Haobo Yang and Brenda Rubenstein. arXiv Preprint. (Preprint)
- Free boundary minimal surfaces in the unit ball with low cohomogeneity, with Peter McGrath and Brian Freidin. Proceedings of the American Mathematical Society (2017). (Journal|Preprint)
- An ab-initio framework for discovering high-temperature superconductors, with Gurgen Melkonyan and Sakthisundar Kasthurirengan. Quantum Studies: Mathematics and Foundations (2017). (Journal)
- Engineering room-temperature superconductors using ab-initio calculations, with Armen Gulian and Gurgen Melkonyan. Physics Proceedia (2015). (Journal)