# Selected Recent Publications (from 130+ in referred journals)

1. A Causality-DeepONet for Causal Responses of Linear Dynamical Systems, Lizuo Liu,  Kamaljyoti Nath, and Wei Cai, arXiv:2209.08397, 17 Sep 2022
2. FBSDE based neural network algorithms for high-dimensional quasilinear parabolic PDEs, Wenzhong Zhang, Wei Cai, arXiv: 2012.07924, (12/2020),  Journal of Computational Physics, 470 (2022) 111557.
3. Linearized Learning Methods with Multiscale Deep Neural Networks for Stationary Navier-Stokes Equations with Oscillatory Solutions, Lizuo Liu, Bo Wang, Wei Cai, arXiv:2012.07924 , 12/2020, submitted to Journal of Computational Physics, 8/2022.
4. A Parallel Iterative Probabilistic Method For Mixed Problems Of Laplace Equations with The Feynman-Kac Formula Of Killed Brownian Motions, Cuiyang Ding, Chanhao Yan, Xuan Zeng, And Wei Cai, to appear in  SISC, 2/2022.
5. Fast multipole method for 3-D Helmholtz equation in layered media, Bo Wang,, Wenzhong Zhang, Wei Cai, arXiv:1902.05132 , SIAM J. Sci. Comput., 41(6), A3954–A3981, (2019). CPU benchmark data (200 Million sources)
6. Exponential Convergence for Multipole Expansion And Translation To Local Expansions For Sources In Layered Media: 2-D Acoustic Wave, Wenzhong Zhang, Bo Wang, Wei Cai, arXiv:1809.07716 (September 20, 2018) , SIAM Numerical Analysis, 58 (3) (2020) 1440-1468.
7. A Matrix Basis Formulation for the Dyadic Green’s Functions of Maxwell’s Equations in Layered Media, Wenzhong Zhang, Bo Wang, Wei Cai, to appear in the SIAM Applied Math, 6/23/2022.
8. Multi-scale Deep Neural Network (MscaleDNN) Methods for Oscillatory Stokes Flows in Complex Domains, Bo Wang, Wenzhong Zhang, Wei Cai,  arXiv:2009.12729, Communications in Computational Physics, Vol. 28, No. 5, pp. 2139-2157, November, 2020.
9. A Highly Scalable Boundary Integral Equation and Walk-On-Spheres (BIE-WOS)Method for the Laplace Equation with Dirichlet Data, Communications in Computational Physics, Vol. 29, No. 5, pp. 1446-1468, May, 2021.
10. A Path Integral Monte Carlo (PIMC) Method based on Feynman-Kac Formula for Electrical Impedance Tomography, Cuiyang Ding, Yijing Zhou, Wei Cai, Xuan Zeng, And Chanhao Yan, submitted to Journal of Computational Physics., October, 2020,  arXiv:1907.13147. 2019.
11. A phase shift deep neural network for high frequency approximation and wave problems , Wei Cai, Xiaoguang Li, Lizuo Liu, arXiv:1909.11759, Sept. 23, 2019, and  SIAM J. Sci. Comput., 2020;42(5):A3285-312.
12. Multi-scale Deep Neural Network (MscaleDNN) for Solving Poisson-Boltzmann Equation in Complex Domains,  Ziqi Liu, Wei Cai, Zhiqin Xu, arxiv July, 2020, Communications in Computational Physics, Vol. 28, No. 5, pp. 1970-2001, November 2020.
13. Multi-scale Deep Neural Networks for Solving High Dimensional PDEs, Wei Cai, Zhiqin John Xu, arXiv:1910.11710 .
14. A high order efficient numerical method for 4-D Wigner equation of quantum double-slit interferences, Z.Z. Chen, S.H. Shao, W. Cai, Journal of Computational Physics, 396 (2019) 54-71.
15. An O(NlogN) hierarchical random compression method for kernel matrices by sampling partial matrix entries, Duan Chen, Wei Cai,  Journal of Computational Physics, 397 (2019) 108828.
16. What is the fractional Laplacian? A comparative review with new results,  Ann Lischke, Guofei Pang, Mamikon Gulian, Fangying Song, Christian Glusa, Xiaoning Zheng,Zhiping Mao, Wei Cai, Mark M. Meerschaert, Mark Ainsworth, George Em Karniadakis, Journal of Computational Physics. 2020 Mar 1;404:109009.
17. A Heterogeneous FMM for Layered Media Helmholtz Equation I: Two Layers in R^2, M.H. Cho, J.F. Huang, D. Chen, W. Cai,  Journal of Computational Physics, 369 (2018) 237–251.
18. A computational stochastic methodology for the design of random meta-materials under geometric constraints, Ivi C Tsantili Min Hyung Cho Wei Cai George Em Karniadakis,  SIAM J. on Scientific computing, Vol. 40, No. 2, pp. B353–B378, 2018.
19. Accurate and Efficient Nystrom Volume  Integral Equation Method for Electromagnetic Scattering of 3-D Meta-materials in Layered Media,  Duan Chen Min Hyung Cho Wei Cai,    SIAM J. on Scientific computing, Vol. 40, No. 1, pp. B259–B282, 2018.
20. Discovering variable fractional orders of advection-dispersion equations from field data using multi-fidelity Bayesian optimization. Pang G, Perdikaris P, Cai W, Karniadakis GE., Journal of Computational Physics. 2017 Nov 1;348(C):694-714.
21. C.H. Yan, W. Cai, X. Zeng , A parallel method for solving Poisson equations with Dirichlet data using local boundary integral equations and random walks, SIAM J. Scientific computing, (2013), vol. 35, No. 4, pp. B868-B889.