Publications

1. Solving high-dimensional inverse problems with auxiliary uncertainty via operator learning with limited data
   with Joseph Hart, Indu Manickam and Laura Swiler.
   Journal of Machine Learning for Modeling and Computing (2023)
   
   
2. Fractional modeling in action: a survey of nonlocal models for subsurface transport, turbulent flows, and anomalous materials
   with Jorge L. Suzuki, Mohsen Zayernouri, and Marta D'Elia.
   Journal of Peridynamics and Nonlocal Modeling (2023)
   
   
3. Error-in-variables modelling for operator learning
   with Ravi G. Patel, Indu Manickam, and Myoungkyu Lee.
   Proceedings of Machine Learning Research (2022)
   
   
4. Connections between nonlocal operators: from vector calculus identities to a fractional Helmholtz decomposition
   with Marta D'Elia, Tadele Mengesha, and James M. Scott.
   Fractional Calculus and Applied Analysis (2022)
   
   
5. Analysis of anisotropic nonlocal diffusion models: well-posedness of fractional problems for anomalous transport
   with Marta D'Elia.
   Numerical Mathematics: Theory, Methods and Applications (2022)
   
   
6. Gaussian processes constrained by boundary value problems
   with Ari Frankel and Laura Swiler.
   Computer Methods in Applied Mechanics and Engineering (2022)
   
   
7. Distribution and pressure of active Levy swimmers under confinement
   with Tingtao Zhou, Zhiwei Peng, and John F. Brady.
   Journal of Physics A: Mathematical and Theoretical (2021)
   
   
8. Partition of unity networks: deep $hp$-approximation
   with Kookjin Lee, Nathaniel A. Trask, Ravi G. Patel, and Eric C. Cyr.
   AAAI-MLPS (2021)
   
   
9. A block coordinate descent optimizer for classification problems exploiting convexity
   with Ravi G. Patel, Nathaniel A. Trask, and Eric C. Cyr.
   AAAI-MLPS (2021)
   
   
10. Towards a unified theory of fractional and nonlocal vector calculus
    with Marta D'Elia, Hayley Olson, and George Em Karniadakis.
    Fractional Calculus and Applied Analysis (2021)
    
    
11. 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)
    
    
12. Data-driven learning of 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)
    
    
13. 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)
    
    
14. 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 (2020)
    
    
15. Machine learning of space-fractional differential equations
    with Maziar Raissi, Paris Perdikaris, and George Karniadakis.
    Journal of Computational Physics (2019)
    
    
16. Free boundary minimal surfaces in the unit ball with low cohomogeneity
    with Peter McGrath and Brian Freidin.
    Proceedings of the American Mathematical Society (2017)
    
    
17. An ab-initio framework for discovering high-temperature superconductors
    with Gurgen Melkonyan and Sakthisundar Kasthurirengan.
    Quantum Studies: Mathematics and Foundations (2017)
    
    
18. Engineering room-temperature superconductors using ab-initio calculations
    with Armen Gulian and Gurgen Melkonyan.
    Physics Proceedia (2015)
    
    

1. Probabilistic partition of unity networks: clustering based deep approximation.
   With Nat Trask, Andy Huang, and Kookjin Lee.
   12 pages
   
   
2. Stochastic solution of elliptic and parabolic boundary value problems for the spectral fractional Laplacian.
   With Guofei Pang.
   45 pages
   
   
3. Fractional Path Integral Monte Carlo.
   With Haobo Yang and Brenda M Rubenstein.
   18 pages
   
   

1. Discovering and Solving Fractional-Order Partial Differential Equations: Machine Learning and Monte Carlo Methods.
   Brown University, January 2019.