Books and Book chapters

4. Z. Zhang and G. E. Karniadakis. Numerical methods for stochastic partial differential equations with white noise, Springer, 2017.

3. A. Lischke, M. Zayernouri, and Z. Zhang. Spectral and spectral element methods for fractional advection-diffusion-reaction equations. Handbook of Fractional Calculus with Applications. Volume 3: Numerical Methods. De Gruyter, 2019

2. H. Yan, Z. Zhang, and J. Zou. Dynamic space-time model for syndromic surveillance with particle filters and Dirichlet process. Proceeding in First International Conference on InfoSymbiotics / DDDAS (Dynamic Data Driven Applications Systems), 2018

1. Z. Zhang, M. Choi, and G. E. Karniadakis. Anchor points matter in ANOVA decomposition. In Spectral and High Order Methods for Partial Differential Equations, pages 347-355. Springer, 2011


1. Ehsan Kharazmi, Zhongqiang Zhang, George EM Karniadakis, VPINNs: Variational Physics-Informed Neural Networks For Solving Partial Differential Equations. PDF at arxiv

2. Y. Dong and Z. Zhang. A half-order numerical scheme for nonlinear SDEs with one-sided Lipschitz drift and Holder continuous diffusion coefficients. Submitted,

Peer-reviewed conference Papers

H. Yan, Z. Zhang and J. Zou. An Online Spatio-Temporal Model for Inference and Predictions of Taxi Demand, Proceedings of IEEE International Conference on Big Data 2017.

Peer-reviewed Journal Papers

Scientific machine learing

3. Lu, Lu; Meng, Xuhui; Cai, Shengze; Mao, Zhiping; Goswami, Somdatta; Zhang, Zhongqiang; Karniadakis, George Em A comprehensive and fair comparison of two neural operators (with practical extensions) based on FAIR data. Comput. Methods Appl. Mech. Engrg. 393 (2022), Paper No. 114778

2. Lu Lu, Pengzhan Jin, Guofei Pang, Zhongqiang Zhang and George Em Karniadakis, Learning nonlinear operators via DeepONet based on the universal approximation theorem of operators, Nature Machine Intelligence volume 3, pages 218-229 (2021) PDF at arxiv

1. Kharazmi, Ehsan; Zhang, Zhongqiang; Karniadakis, George E. M. hp-VPINNs: variational physics-informed neural networks with domain decomposition. Comput. Methods Appl. Mech. Engrg. 374 (2021), Paper No. 113547 PDF ar arxiv

Numerical space-fractional differential equations

10. Caiyu Jiao, Changpin Li, Hexiang Wang, Zhongqiang Zhang, A modified walk-on-sphere method for high dimensional fractional Poisson equation, Numerical methods for partial differential equations, 2022, arxiv

9. Cao, Wanrong; Hao, Zhaopeng; Zhang, Zhongqiang Optimal strong convergence of finite element methods for one-dimensional stochastic elliptic equations with fractional noise. J. Sci. Comput. 91 (2022), no. 1, Paper No. 1

8. Hao, Zhaopeng; Zhang, Zhongqiang Numerical approximation of optimal convergence for fractional elliptic equations with additive fractional Gaussian noise. SIAM/ASA J. Uncertain. Quantif. 9 (2021), no. 3, 1013--1033.

7. F. Wang, Z. Zhang, Z. Zhou, A spectral Galerkin approximation of optimal control problem governed by fractional advection-diffusion-reaction equations, J. Comput. Appl. Math., 386, (2021), 113233. Full text

6. Hao, Zhaopeng; Zhang, Zhongqiang Fast spectral Petrov-Galerkin method for fractional elliptic equations. Appl. Numer. Math. 162 (2021), 318-330.

5. Hao, Zhaopeng; Li, Huiyuan; Zhang, Zhimin; Zhang, Zhongqiang Sharp error estimates of a spectral Galerkin method for a diffusion-reaction equation with integral fractional Laplacian on a disk. Math. Comp. 90 (2021), no. 331, 2107-2135.

4. Z. Hao and R. Du and Z. Zhang, Fractional centered difference scheme for high-dimensional integral fractional Laplacian, J. Comput. Phys., Article no. 109851, 2020. Full text

3. Z. Hao, G. Lin, Z. Zhang. Error estimates of a spectral Petrov-Galerkin method for two-sided fractional reaction-diffusion equations. Appl. Math. Comp., 374, 125045, 2020 Full text

2. Z. Hao and Z. Zhang. Optimal regularity and error estimates of a spectral Galerkin method for fractional advection-diffusion- reaction equations. SIAM J. NUMER. ANAL., 58(1), 211--233, 2020. Full text

1. Z. Zhang. Error estimates of spectral Galerkin methods for a linear fractional reaction-diffusion equation, Journal of Scientific Computing, 78, 1087-1110, 2019 Full text at Springer

Data-driven problems and statistics

30. Y. Dong, M. Humi and Z. Zhang, Satellite orbits and fractional mechanics. Rom. J. Tech. Sci. Appl. Mech. 65 (2020), no. 3, 181-196.

29. J. Zou, Z. Zhang and H. Yan. Hybrid Hierarchical Bayesian Model for Spatio-Temporal Surveillance Data, Statistics in Medicine, Statistics in Medicine, 2018 Full text at Wiley

28. Z. Zhang and X. Yang and Guang Lin. POD-based constrained sensor placement and field reconstruction from noisy wind measurements: A perturbation study. Mathematics, 4(2):26, 2016. Full text

Numerical time-fractional differential equations

27. Z. Yang, X. Zheng, Z. Zhang, H. Wang, Strong convergence of a Euler-Maruyama scheme to a variable-order fractional stochastic differential equation driven by a multiplicative white noise, Chaos, Solitions & Fractals, (2020), 110392 Full text

26. Xiangcheng Zheng and ZhongqiangZhang and Hong Wang, Analysis of a nonlinear variable-order fractional stochastic differential equation, Applied Mathematics Letters, Article no. 106461, 2020. Full text

25. F. Zeng, Z. Zhang, and G. E. Karniadakis. Second-order numerical methods for multi-term fractional differential equations: Smooth and non-smooth solutions. Computer Methods in Applied Mechanics and Engineering 327, 478-502, 2017. PDF at arXiv

24. W. Cao, F. Zeng, Z. Zhang, and G. E. Karniadakis. Implicit-explicit difference schemes for nonlinear fractional differential equations with non-smooth solutions. SIAM J. Sci. Comput., 38(5), A3070-A3093, 2016. PDF at arXiv

23. F. Zeng, Z. Zhang, and G. E. Karniadakis. Fast difference schemes for solving high-dimensional time-fractional subdiffusion equations, J. Comput. Phys., 307:15-33, 2016.

22. F. Zeng, Z. Zhang and G. E. Karniadakis. A generalized spectral collocation method with tunable accuracy for variable-order fractional differential equations, SIAM J. Sci. Comput., 37(6):A2710-A2732, 2015.

21. W.-R. Cao, Z. Zhang and G. E. Karniadakis. Time-Splitting Schemes for Fractional Differential Equations I: Smooth Solutions, SIAM J. Sci. Comput., 37(4): A1752-A1776, 2015. PDF

20. Z. Zhang, F. Zeng, and G. E. Karniadakis. Optimal error estimates of spectral Petrov-Galerkin and collocation methods for initial value problems of fractional differential equations. SIAM J. Numer. Anal., 53(4):2074-2096, 2015. PDF

19. M. Zayernouri, W. Cao, Z. Zhang, and G. E. Karniadakis. Spectral and discontinuous spectral element methods for fractional delay equations. SIAM J. Sci. Comput., 36(6):B904-B929, 2014.

Numerical stochastic differential equations

18. Z. Zhang and H. Ma. Order-preserving strong schemes for SDEs with locally lipschitz coefficients. Appl. Numer. Math., 122:1-16, 2017. PDF at arXiv

17. Z. Zhang, B. Rozovskii, and G. E. Karniadakis. Strong and weak convergence order of finite element methods for stochastic PDEs with spatial white noise, Numer. Math., 134(1):61-89, 2016. Full text at Springer

16. W.-R. Cao, Z. Zhang, and G. E. Karniadakis. Numerical methods for stochastic delay differential equations via Wong-Zakai approximation. SIAM J. Sci. Comput., 37(1): A295-A318, 2015. PDF

15. Z. Zhang, M. V. Tretyakov, B. Rozovskii, and G. E. Karniadakis. Wiener chaos vs stochastic collocation methods for linear advection-diffusion equations with multiplicative white noise. SIAM J. Numer. Anal., 53(1): 153-183, 2015. PDF

14. Z. Zhang, M. V. Tretyakov, B. Rozovskii, and G. E. Karniadakis. A recursive sparse grid collocation method for differential equations with white noise. SIAM J. Sci. Comput., 36(4):A1652-A1677, 2014. PDF

13. W. Cao, P. Hao, and Z. Zhang. Split-step theta-method for stochastic delay differential equations. Appl. Numer. Math., 76:19-33, 2014.

12. M. V. Tretyakov and Z. Zhang. A fundamental mean-square convergence theorem for SDEs with locally Lipschitz coefficients and its applications. SIAM J. Numer. Anal., 51(6):3135-3162, 2013. PDF at arXiv

11. Z. Zhang, X. Yang, G. Lin, and G. E. Karniadakis. Numerical solution of the Stratonovich- and Ito-Euler equations: application to the stochastic piston problem. J. Comput. Phys., 236:15-27, 2013.

10. W. Cao and Z. Zhang. On exponential mean-square stability of two-step Maruyama methods for stochastic delay differential equations. J. Comput. Appl. Math., 245:182-193, 2013.

9. W. Cao and Z. Zhang. Simulations of two-step Maruyama methods for nonlinear stochastic delay differential equations. Adv. Appl. Math. Mech., 4(6):821-832, 2012.

8. Z. Zhang, B. Rozovskii, M. V. Tretyakov, and G. E. Karniadakis. A multistage Wiener chaos expansion method for stochastic advection-diffusion-reaction equations. SIAM J. Sci. Comput., 34(2):A914-A936, 2012.

7. Z. Zhang, M. Choi, and G. E. Karniadakis. Error estimates for the ANOVA method with polynomial chaos interpolation: tensor product functions. SIAM J. Sci. Comput., 34(2):A1165-A1186, 2012.

Spectral methods

6. L. Chen, Z. Zhang, and H. Ma. The dissipative spectral methods for the first-order linear hyperbolic equations. Numer. Math. Theor. Meth. Appl., 5(3):493-508, 2012.

5. T. Shen, Z. Zhang, and H. Ma. Optimal error estimates of the Legendre tau method for second-order differential equations. J. Sci. Comput., 42(2):198-215, 2010. Abstract

4. R. Ju, H. Ma, and Z. Zhang. Multisymplectic Fourier pseudospectral method for the MKdV equation. Commun. Appl. Math. Comput., 23(1):81-86, 2009.

3. W. Guo, Z. Zhang, and H. Ma. Optimal error estiamtes of the multi-symplectic Fourier pseudospectral method for a nonlinear Schrodinger equation. J. Shanghai Univ. Nat. Sci., 15(5):487-492, 2009.

2. Z. Zhang and H. Ma. A rational spectral method for the KdV equation on the half line. J. Comput. Appl. Math., 230(2):614-625, 2009.

1. Z. Zhang and H. Ma. A combined modified rational spectral method for the BBM equation on the half-line. Commun. Appl. Math. Comput., 21(1):66-76, 2007.