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SOC-MartNet: A Martingale Neural Network for the Hamilton-Jacobi-Bellman Equation without Explicit $\inf_{u \in U} H$ in Stochastic Optimal Controls , Cai W, Fang S, Zhou T., arXiv preprint arXiv:2405.03169. 2024 May 6.

DeepMartNet–A Martingale Based Deep Neural Network Learning Method for Dirichlet BVPs and Eigenvalue Problems of Elliptic PDEs in R^d . Cai W, He A, Margolis D. arXiv preprint arXiv:2311.09456. 2023 Nov 15

On Spectral Bias Reduction of Multi-scale Neural Networks for Regression and Boundary Value Problems, Bo Wang, Heng Yuan, Lizuo Liu, Wenzhong Zhang, Wei Cai, updated 8/29/2023, first version in arXiv preprint arXiv:2212.03416. 2022 Dec 7..

A Causality-DeepONet for Causal Responses of Linear Dynamical Systems , Lizuo Liu, Kamaljyoti Nath, and Wei Cai, Commun. Comput. Phys. , Vol. 35, No. 5, pp. 1194-1228, May, 2024.

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, Journal of Computational Physics. , 476 (2023) 111862.

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.

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, EAJAM Vol. 13, No. 3, pp. 740-758, August 2023.

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, SIAM Scientific Computing, 44 , A3413-A3435, 2022.

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 )

Fast multipole method for 3-D Laplace equation in layered media . B. Wang, W. Zhang, W. Cai, Computer Physics Communications. 259:107645, (2021).

Fast multipole method for 3-D Poisson-Boltzmann equation in layered electrolyte-dielectric media . Wang, W. Zhang, W. Cai, Journal of Computational Physics . (pdf) 439:110379 (2021).

Exponential Convergence Theory of the Multipole and Local Expansions for the 3-D Laplace Equation in Layered Media , Wenzhong Zhang, Bo Wang and Wei Cai, Annals of Applied Mathematics , 2023, in press.

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.

A Matrix Basis Formulation for the Dyadic Green’s Functions of Maxwell’s Equations in Layered Media , Wenzhong Zhang, Bo Wang, Wei Cai, SIAM Applied Math, 82 (5), 1710-1732, 2022.

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.

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.

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.

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. A correction (CiCP, 2023) .

Multi-scale Deep Neural Networks for Solving High Dimensional PDEs, Wei Cai, Zhiqin John Xu, arXiv:1910.11710 .

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.

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.

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.

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.

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.

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.

Computation of Local Time of Reflecting Brownian Motion and Probabilistic Representation of the Neumann Problem , , Communications in Mathematical Sciences, Y.J. Zhou, W. Cai, E. Hsu, Vol. 15, No. 1, pp. 237–259, 2017.

High order hierarchical divergence-free constrained transport H (div) finite element method for magnetic induction equation . Cai W, Hu J, Zhang S. Numerical Mathematics: Theory, Methods and Applications. 2017 May;10(2):243-54.

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.

ICSM: An order N method for calculating electrostatic interactions added to TINKER , K. Baker, A. Baumketner, Y.C. Lin, S.Z. Deng, D.Jacobs, W. Cai, Computer Physics Communication, ,Vo 184, Issue 1, January 2013, Pages 19–26.

Linear scaling discontinuous Galerkin density matrix minimization method with local orbital enriched finite element basis: 1-D lattice model system , T. Lu, W. Cai, J.G. Xin, Y. L. Guo, Communication in computational physics, vol. 14, No. 2, (2013), pp. 276-300.

On the Construction of Well-Conditioned Hierarchical Bases for H(div)-Conforming ℝn Simplicial Elements . J. Xin, W. Cai, N. Guo, Communications in Computational Physics . 2013 Sep;14(3):621-38.

Divergence-free H(div)-conforming hierarchical bases for Magnetohydrodynamics (MHD) , Wei Cai, Jian Wu and Jianguo Xin, Communications in Mathematics and Statistics (2013) 1:19-35.

Well-conditioned Orthonormal Hierarchical Bases on ℝ n Simplicial Elements . Xin, W. Cai, Journal of scientific computing . 2012 Feb;50(2):446-61.

On the construction of well-conditioned hierarchical bases for tetrahedral H (curl)-conforming Nédélec element . J. Xin, N. Guo N, W. Cai, Journal of Computational Mathematics. 2011 Sep 1:526-42.

Fast analytical methods for macroscopic electrostatic models in biomolecular simulations , Z. L. Xu, W. Cai, SIAM Rev iew 53, pp. 683-720, 2012.

Boundary Treatments in the Nonequilibrium Green’s functions (NEGF) methods for transport in Nano-MOSFETs , H. Y. Jiang, S.H. Shao, W. Cai, P.W. Zhang, the Journal of Computational Physics, 227, (2008), pp. 6553

Extending the Fast Multipole Method to Charges Inside or Outside a Dielectric Sphere , W. Cai, S. Z. Deng, D. Jacobs, the Journal of Computational Physics, 223 (2007), pp. 846-864.

Discontinuous Galerkin Methods for Dispersive and Lossy Maxwell’s Equations and PML Boundary Conditions , T. Lu, P.W. Zhang, W. Cai, Journal of Computational Physics, 200 (2004), pp. 549-580.

An Upwinding Embedded Boundary Methods for Maxwell equations , W. Cai, S. Deng, Journal of Computational Physics, 190, 2003.

High Order mixed RWG basis functions for electromagnetic applications , IEEE transaction on Microwave Theory and Technology, W. Cai, T. J. Yu, H. Wang, Y. J. Yu, Vol. 49, No. 7, pp. 1295-1303, July 2001.

Fast calculation of dyadic Green’s functions for electromagnetic scattering in a multi-layered medium , Cai, T.J. Yu, Journal of Computational Physics, 165, 1-21, 2000.

Adaptive Multiresolution Collocation Method for the Initial Value Boundary Problem of Nonlinear PDE’s , W. Cai, J.Z. Wang, SIAM Journal of Numerical Analysis, Volume 33, Number 3, pp. 937-970, June 1996.

Essentially nonoscillatory spectral Fourier methods for shock wave calculations . W. Cai, D. Gottlieb, CW. Shu, Mathematics of Computation. 1989;52(186):389-410.

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