Optimal Error Estimates of the Legendre Tau Method for Second-Order Differential Equations. - Ting-Ting Shen; Zhongqiang Zhang 003; He-Ping Ma (J. Sci. Comput., 2010)

Error Estimates for the ANOVA Method with Polynomial Chaos Interpolation: Tensor Product Functions. - Zhongqiang Zhang 003; Minseok Choi; George E. Karniadakis (SIAM J. Sci. Comput., 2012)

A Multistage Wiener Chaos Expansion Method for Stochastic Advection-Diffusion-Reaction Equations. - Zhongqiang Zhang 003; Boris Rozovskii; Michael V. Tretyakov; George E. Karniadakis (SIAM J. Sci. Comput., 2012)

On exponential mean-square stability of two-step Maruyama methods for stochastic delay differential equations. - Wanrong Cao; Zhongqiang Zhang 003 (J. Comput. Appl. Math., 2013)

Numerical solution of the Stratonovich- and Ito-Euler equations: Application to the stochastic piston problem. - Zhongqiang Zhang 003; Xiu Yang; Guang Lin 001; George E. Karniadakis (J. Comput. Phys., 2013)

A Fundamental Mean-Square Convergence Theorem for SDEs with Locally Lipschitz Coefficients and Its Applications. - Michael V. Tretyakov; Zhongqiang Zhang 003 (SIAM J. Numer. Anal., 2013)

Spectral and Discontinuous Spectral Element Methods for Fractional Delay Equations. - Mohsen Zayernouri; Wanrong Cao; Zhongqiang Zhang 003; George E. Karniadakis (SIAM J. Sci. Comput., 2014)

A Recursive Sparse Grid Collocation Method for Differential Equations with White Noise. - Zhongqiang Zhang 003; Michael V. Tretyakov; Boris Rozovskii; George E. Karniadakis (SIAM J. Sci. Comput., 2014)

Wiener Chaos Versus Stochastic Collocation Methods for Linear Advection-Diffusion-Reaction Equations with Multiplicative White Noise. - Zhongqiang Zhang 003; Michael V. Tretyakov; Boris Rozovskii; George E. Karniadakis (SIAM J. Numer. Anal., 2015)

Optimal Error Estimates of Spectral Petrov-Galerkin and Collocation Methods for Initial Value Problems of Fractional Differential Equations. - Zhongqiang Zhang 003; Fanhai Zeng; George E. Karniadakis (SIAM J. Numer. Anal., 2015)

Numerical Methods for Stochastic Delay Differential Equations Via the Wong-Zakai Approximation. - Wanrong Cao; Zhongqiang Zhang 003; George E. Karniadakis (SIAM J. Sci. Comput., 2015)

Time-Splitting Schemes for Fractional Differential Equations I: Smooth Solutions. - Wanrong Cao; Zhongqiang Zhang 003; George E. Karniadakis (SIAM J. Sci. Comput., 2015)

A Generalized Spectral Collocation Method with Tunable Accuracy for Variable-Order Fractional Differential Equations. - Fanhai Zeng; Zhongqiang Zhang 003; George E. Karniadakis (SIAM J. Sci. Comput., 2015)

Fast difference schemes for solving high-dimensional time-fractional subdiffusion equations. - Fanhai Zeng; Zhongqiang Zhang 003; George E. Karniadakis (J. Comput. Phys., 2016)

Strong and weak convergence order of finite element methods for stochastic PDEs with spatial white noise. - Zhongqiang Zhang 003; Boris Rozovskii; George E. Karniadakis (Numerische Mathematik, 2016)

Implicit-Explicit Difference Schemes for Nonlinear Fractional Differential Equations with Nonsmooth Solutions. - Wanrong Cao; Fanhai Zeng; Zhongqiang Zhang 003; George E. Karniadakis (SIAM J. Sci. Comput., 2016)

An online spatio-temporal model for inference and predictions of taxi demand. - Hong Yan; Zhongqiang Zhang 003; Jian Zou (IEEE BigData, 2017)

Dynamic Space-Time Model for Syndromic Surveillance with Particle Filters and Dirichlet Process. - Hong Yan; Zhongqiang Zhang 003; Jian Zou (Handbook of Dynamic Data Driven Applications Systems, 2018)

Variational Physics-Informed Neural Networks For Solving Partial Differential Equations. - Ehsan Kharazmi; Zhongqiang Zhang 003; George Em Karniadakis (CoRR, 2019)

Error Estimates of Spectral Galerkin Methods for a Linear Fractional Reaction-Diffusion Equation. - Zhongqiang Zhang 003 (J. Sci. Comput., 2019)

Error estimates of a spectral Petrov-Galerkin method for two-sided fractional reaction-diffusion equations. - Zhaopeng Hao; Guang Lin 001; Zhongqiang Zhang 003 (Appl. Math. Comput., 2020)

Analysis of a nonlinear variable-order fractional stochastic differential equation. - Xiangcheng Zheng; Zhongqiang Zhang 003; Hong Wang 018 (Appl. Math. Lett., 2020)

hp-VPINNs: Variational Physics-Informed Neural Networks With Domain Decomposition. - Ehsan Kharazmi; Zhongqiang Zhang 003; George Em Karniadakis (CoRR, 2020)

Error estimates of residual minimization using neural networks for linear PDEs. - Yeonjong Shin; Zhongqiang Zhang 003; George Em Karniadakis (CoRR, 2020)

Optimal Regularity and Error Estimates of a Spectral Galerkin Method for Fractional Advection-Diffusion-Reaction Equations. - Zhaopeng Hao; Zhongqiang Zhang 003 (SIAM J. Numer. Anal., 2020)

Convergence rate of DeepONets for learning operators arising from advection-diffusion equations. - Beichuan Deng; Yeonjong Shin; Lu Lu 010; Zhongqiang Zhang 003; George Em Karniadakis (CoRR, 2021)

A spectral Galerkin approximation of optimal control problem governed by fractional advection-diffusion-reaction equations. - Fangyuan Wang; Zhongqiang Zhang 003; Zhaojie Zhou (J. Comput. Appl. Math., 2021)

Fractional centered difference scheme for high-dimensional integral fractional Laplacian. - Zhaopeng Hao; Zhongqiang Zhang 003; Rui Du (J. Comput. Phys., 2021)

Numerical Approximation of Optimal Convergence for Fractional Elliptic Equations with Additive Fractional Gaussian Noise. - Zhaopeng Hao; Zhongqiang Zhang 003 (SIAM/ASA J. Uncertain. Quantification, 2021)

Sharp error estimates of a spectral Galerkin method for a diffusion-reaction equation with integral fractional Laplacian on a disk. - Zhaopeng Hao; Huiyuan Li; Zhimin Zhang 002; Zhongqiang Zhang 003 (Math. Comput., 2021)

Learning nonlinear operators via DeepONet based on the universal approximation theorem of operators. - Lu Lu 010; Pengzhan Jin; Guofei Pang; Zhongqiang Zhang 003; George Em Karniadakis (Nat. Mach. Intell., 2021)

A Modified Walk-on-sphere Method for High Dimensional Fractional Poisson Equation. - Caiyu Jiao; Changpin Li; Hexiang Wang; Zhongqiang Zhang 003 (CoRR, 2022)

Optimal Strong Convergence of Finite Element Methods for One-Dimensional Stochastic Elliptic Equations with Fractional Noise. - Wanrong Cao; Zhaopeng Hao; Zhongqiang Zhang 003 (J. Sci. Comput., 2022)

Approximation rates of DeepONets for learning operators arising from advection-diffusion equations. - Beichuan Deng; Yeonjong Shin; Lu Lu 010; Zhongqiang Zhang 003; George Em Karniadakis (Neural Networks, 2022)

Peridynamic Neural Operators: A Data-Driven Nonlocal Constitutive Model for Complex Material Responses. - Siavash Jafarzadeh; Stewart Silling; Ning Liu; Zhongqiang Zhang 003; Yue Yu (CoRR, 2024)

Two-scale Neural Networks for Partial Differential Equations with Small Parameters. - Qiao Zhuang; Chris Ziyi Yao; Zhongqiang Zhang 003; George Em Karniadakis (CoRR, 2024)

Transformers as Neural Operators for Solutions of Differential Equations with Finite Regularity. - Benjamin Shih; Ahmad Peyvan; Zhongqiang Zhang 003; George Em Karniadakis (CoRR, 2024)

A simple and fast finite difference method for the integral fractional Laplacian of variable order. - Zhaopeng Hao; Siyuan Shi; Zhongqiang Zhang 003; Rui Du (CoRR, 2024)

Score-fPINN: Fractional Score-Based Physics-Informed Neural Networks for High-Dimensional Fokker-Planck-Levy Equations. - Zheyuan Hu 002; Zhongqiang Zhang 003; George Em Karniadakis; Kenji Kawaguchi (CoRR, 2024)

Tackling the Curse of Dimensionality in Fractional and Tempered Fractional PDEs with Physics-Informed Neural Networks. - Zheyuan Hu 002; Kenji Kawaguchi; Zhongqiang Zhang 003; George Em Karniadakis (CoRR, 2024)

On one-step numerical schemes of weak convergence for SDEs with super-linear coefficients. - Yuying Zhao; Xiaojie Wang 007; Zhongqiang Zhang 003 (CoRR, 2024)

Fractional-order dependent Radial basis functions meshless methods for the integral fractional Laplacian. - Zhaopeng Hao; Zhiqiang Cai 001; Zhongqiang Zhang 003 (CoRR, 2024)

Collocation methods for integral fractional Laplacian and fractional PDEs based on radial basis functions. - Qiao Zhuang; Alfa R. H. Heryudono; Fanhai Zeng; Zhongqiang Zhang 003 (Appl. Math. Comput., 2024)

Jacobi spectral collocation method of space-fractional Navier-Stokes equations. - Yujian Jiao; Tingting Li; Zhongqiang Zhang 003 (Appl. Math. Comput., 2025)

Tensor neural networks for high-dimensional Fokker-Planck equations. - Taorui Wang; Zheyuan Hu 002; Kenji Kawaguchi; Zhongqiang Zhang 003; George Em Karniadakis (Neural Networks, 2025)

Score-Based Physics-Informed Neural Networks for High-Dimensional Fokker-Planck Equations. - Zheyuan Hu 002; Zhongqiang Zhang 003; George E. Karniadakis; Kenji Kawaguchi (SIAM J. Sci. Comput., 2025)

Monotone Peridynamic Neural Operator for Nonlinear Material Modeling with Conditionally Unique Solutions. - Jihong Wang; Xiaochuan Tian; Zhongqiang Zhang 003; Stewart Silling; Siavash Jafarzadeh; Yue Yu (CoRR, 2025)

A Spectral Galerkin Method for the Mixed Local and Nonlocal Elliptic Equations. - Zhaopeng Hao; Zhongqiang Zhang 003 (J. Sci. Comput., 2025)

Weak Error Analysis for Strong Approximation Schemes of SDEs with Super-Linear Coefficients II Finitely Many Bounded Moments and Higher-Order Schemes. - Yuying Zhao; Xiaojie Wang 007; Zhongqiang Zhang 003 (J. Sci. Comput., 2025)