**(1) Neural Networks and Scientific Machine Learning**

- (with M. Liu)
*Self-adaptive ReLU neural network method in least-squares data fitting*,*Principles and Applications of Adaptive Artificial Intelligence*, Chapter 11 (2024), 242-262. DOI: 10.4018/979-8-3693-0230-9.ch011. - (with J. Chen and M. Liu)
*Self-adaptive deep neural network: numerical approximation to functions and PDEs*,*J. Comput. Phys.*, 455 (2022), 111021. - (with M. Liu)
*Adaptive two-layer ReLU neural network: II. RITZ approximation to elliptic PDEs*,*Comput. Math. Appl.*, 113 (2022), 103-116. - (with M. Liu and J. Chen)
*Adaptive two-layer ReLU neural network: I. Best least-squares approximation*,*Comput. Math. Appl.*, 113 (2022), 34-44. - (with J. Choi and M. Liu)
*Least-squares neural network (LSNN) method for linear advection-reaction equation: non-constant jumps*,*Int'l J. Numer. Anal. Modeling*, to appear. - (with J. Choi and M. Liu)
*Least-squares neural network (LSNN) method for linear advection-reaction equation: general discontinuous interface*,*arXiv:2301.06156v4[math.NA]*,*SIAM J. Sci. Comput.*, in revision. - (with J. Chen and M. Liu)
*Least-squares neural network (LSNN) method for scalar nonlinear hyperbolic conservation laws: discrete divergence operator*,*arXiv2110.10895v3[math.NA]*,*J. Comput. Appl. Math.*, 433 (2023) 115298. - (with J. Chen and M. Liu)
*Least-squares ReLU neural network (LSNN) method for scalar nonlinear hyperbolic conservation law*,*Appl. Numer. Math.*, 174 (2022), 163-176. - (with J. Chen and M. Liu)
*Least-squares ReLU neural network (LSNN) method for linear advection-reaction equation*,*J. Comput. Phys.*, 443 (2021), 110514. - (with J. Chen, M. Liu, and X. Liu)
*Deep least-squares methods: an unsupervised learning-based numerical method for solving elliptic PDEs*,*J. Comput. Phys.*, 420 (2020), 109707. - (with B. Hejnal)
*Evolving neural network (ENN) method for one-dimensional scalar hyperbolic conservation laws*,*SIAM J. Sci. Comput.*, submitted,*arXiv:2312.06919[math.NA]* - (with T. Ding, M. Liu, X. Liu, and J. Xia)
*A structure-guided Gauss-Newton method for shallow ReLU neural network*, submitted,*arXiv:2404.05064[cs.LG]* - (with A. Dokotorova, R. Falgout, and C. Herrera)
*Fast Iterative solver for neural network method: I. 1D diffusion problems*, submitted.*arXiv:2404.17750[math.NA]* - (with M. Liu and K. Ramani)
*Deep Ritz method with adaptive quadrature for linear elasticity*,*Comput. Methods in Appl. Mech. Eng.*,*415 (2023) 116229*. - (with Z. Hao and M. Park)
*Neural network method for integral fractional Laplace equations*,*East Asian Journal on Applied Mathematics*, 13:1 (2023), 95-118. *(with D. Jiao and M. Liu)**Minimization formulation for neural network based solution of Maxwell's equations in frequency domain*,*in the IEEE International Symposium on Antennas & Propagation, USNC-URSI Radio Science Meeting*, July 10-15, 2022, Denver, USA.- (with M. Liu and D. Jiao)
*Ritz neural network (RitzNN) method for H(curl) problems*,*in the Applied Computational Electromagnetics Society (ACES) Virtual Conference*, August 1-5, 2021, ACES2021OL-1302.

**Adaptive Neural Network (ANE) Method**

**Least-Squares Neural Network (LSNN) Method**

**Evolving Neural Network (ENN) Method**

**Iterative/Optimization/Training Method**

**Deep Ritz Method**

- (with S. Du)
*Adaptive finite element method for Dirichlet boundary control of elliptic partial differential equations*,*J. Sci. Comput.*, 89 (2021), 36. - (with C. He and S. Zhang)
*Generalized Prager-Synge identity and robust equilibrated error estimators for discontinuous elements*,*J. Comput. Appl. Math.*, 398 (2021), 113673. - (with C. He and S. Zhang)
*Improved ZZ error estimators for diffusion problems: discontinuous elements*,*Appl. Numer. Math.*, 159 (2021), 174-189. - (with B. Chen and J. Yang)
*Adaptive least-squares methods for convection-dominant diffusion-reaction problems*, submitted. - (with D. Cai and S. Zhang)
*Robust equilibrated error estimator for diffusion problems: mixed finite elements in two dimensions*,*J. Sci. Cimput.*, 83:22 (2020). - (with D. Cai and S. Zhang)
*Robust equilibrated a posteriori error estimator for higher order finite element approximations to diffusion problems*,*Numer. Math.*, 144:1 (2020) 1-21. - (with D. Cai)
*A hybrid a posteriori error estimator for conforming finite element approximations*,*Comput. Methods in Appl. Mech. Eng.*,339 (2018), 320-340. - (with C. He and S. Zhang)
*Discontinuous finite element methods for interface problems: robust a priori and a posteriori error estimates*,*SIAM J. Numer. Anal.*, 55:1 (2017), 400-418. - (with C. He and S. Zhang)
*Improved ZZ error estimators for diffusion problems: conforming linear elements*,*Comput. Methods in Appl. Mech. Eng.*, 313 (2017), 433-449. - (with C. He and S. Zhang)
*Residual-based a posteriori error estimate for interface problems: nonconforming linear elements*,*Math. Comp.*, 86:304 (2017), 617-636. - (with S. Cao and R. Falgout)
*Robust a posteriori error estimation for finite element approximation to H(curl) problems*,*Comput. Methods in Appl. Mech. Eng.*, 309 (2016), 182-201. - (with S. Cao)
*A recovery-based a posteriori error estimator for H(curl) interface problems*,*Comput. Methods in Appl. Mech. Eng.*, 296 (2015), 169-195. - (with V. Carey, J. Ku, and E.-J. Park)
*Asymptotically exact a posteriori error estimators for first-order div least-squares methods in local and global L2 norm*,*Comput. Math. Appl.*, 70 (2015), 648-659. - (with L. Song and Y. Hou)
*Recovery-based error estimator for stabilized finite element methods for the Stokes equations*,*Comput. Methods in Appl. Mech. Eng.*, 272 (2014), 1-16. - (with S. Zhang)
*Robust equilibrated residual error estimator for diffusion problems: Conforming elements*,*SIAM J. Numer. Anal.*, 50:1 (2012), 151-170. - (with J. Ku)
*Goal-oriendted local a posteriori error estimator for H(div) least-squares finite element method*,*SIAM J. Numer. Anal.*, 49:6 (2011), 2564-2575. - (with S. Zhang)
*Robust residual- and recovery-based a posteriori error estimators for interface problems with flux jumps*,*Numer. Methods Part. Diff. Eqs.*,**28:2**(2012), 476-491 (published online 10/25/2010). - (with S. Zhang)
*Flux recovery and a posteriori error estimators: conforming elements for scalar elliptic equations*,*SIAM J. Numer. Anal.*,**48:2**(2010), 578-602. - (with S. Zhang)
*Recovery-based error estimators for interface problems: mixed and nonconforming finite elements*,*SIAM J. Numer. Anal.*,**48:1**(2010), 30-52. - (with S. Zhang)
*Recovery-based error estimators for interface problems: conforming linear elements*,*SIAM J. Numer. Anal.*,**47:3**(2009), 2132-2156. - (with S. Zhang)
*Recovery-based error estimators for interface problems: mixed and nonconforming finite elements (extended version)*, manuscript, 2008. - (with F. Le Gland and H. Zhang)
*An adaptive local refinement techniques for nonlinear filtering*, Publication Interne No. 954, IRISA, France, October 1995.

- (with J. Ku)
*A dual finite element method for a singularly perturbed reaction-diffusion problem*,*SIAM J. Numer. Anal.*, 58:3 (2020) 1654-1673. - (with F. Bertrand and E.-Y. Park)
*Least-squares methods for elasticity and Stokes equations with weakly imposed symmetry*,*Comput. Methods Appl. Math.*, 19:3 (2019), 415-430. - (with B. Chen)
*Least squares method for the Oseen equation*,*Numer. Methods Part. Diff. Eqs.*, 32:4 (2016), 1289-1303. - (with R. Falgout and S. Zhang)
*Div first-order system LL* (FOSLL*) for second-order elliptic partial differential equations*,*SIAM J. Numer. Anal.*, 53:1 (2015), 405-420. - (with Ku)
*Optimal error estimates for the div least-squares method with data f in L2 and application to nonlinear problems*,*SIAM J. Numer. Anal.*,**47:6**(2010), 4098-4111. - (with F. Attia and G. Starke)
*First-order system least squares for the Signorini contact problem in linear elasticity*,*SIAM J. Numer. Anal.*,**47:4**(2009), 3027-3043. - (with C. Westphal)
*An adaptive mixed least-squares finite element method for viscoelastic fluids of Oldroyd type*,*J. Non-Newt. Fluid Mech.*,**159:1-3**(2009), 72-80. - (with C. Westphal)
*A weighted H(div) least-squares method for second-order elliptic equations*,*SIAM J. Numer. Anal.*,**46:3**(2008), 1640-1651. - (with J. Ku)
*The L2 norm error estimates for the Div least-squares method*,*SIAM J. Numer. Anal.*,**44:4**(2006), 1721-1734. - (with J. Korsawe and G. Starke)
*Adaptive least squares mixed finite element computations for the stress-displacement formulation of linear elasticity*,*Numer. Methods Part. Diff. Eqs.*,**21:1**(2005), 132-148. - (with B. Lee and P. Wang)
*Least-squares methods for incompressible Newtonian fluid flow: linear stationary problems*,*SIAM J. Numer. Anal.*,**42:2**(2004), 843-859. - (with G. Starke)
*Least-squares methods for linear elasticity*,*SIAM J. Numer. Anal.*,**42:2**(2004), 826-842. - (with G. Starke)
*First-order system least squares for the stress-displacement formulation: linear elasticity*,*SIAM J. Numer. Anal.*,**41:2**(2003), 715-730. - (with B.-C. Shin)
*The discrete first-order system least squares: the second-order elliptic boundary value problem*,*SIAM J. Numer. Anal.*,**40:1**(2002), 307-318. - (with T. Manteuffel, S. McCormick, and J. Ruge)
*First-order system LL* (FOSLL*): scalar elliptic partial differential equations*,*SIAM J. Numer. Anal.*,**39:4**(2001), 1418-1445. *Least squares for the perturbed Stokes equations and the Reissner-Mindlin plate*,*SIAM J. Numer. Anal.*,**38:5**(2000), 1561--1581.- (with C.-O. Lee, T. Manteuffel, and S. McCormick)
*First-order system least squares for the Stokes and linear elasticity equations: further results*,*SIAM J. Sci. Comput.*,**21:5**(2000), 1728--1739. - (with C.-O. Lee, T. Manteuffel, and S. McCormick)
*First-order system least squares for linear elasticity: numerical results*,*SIAM J. Sci. Comput.*,**21:5**(2000), 1706--1727. - (with X. Ye)
*Least-squares finite element approximations for compressible Stokes equation*,*Numer. Methods Part. Diff. Eqs,*, 62-70.**16**(2000) - (with X. Ye and H. Zhang)
*Least-squares finite element approximations for the Reissner-Mindlin plate*,*Numer. Lin. Alg. Appl.*,, 479-496.**6:6**(1999) - (with P. Bochev, T. Manteuffel, and S. McCormick)
*Analysis of velocity-flux least-squares principles for Navier-Stokes equations: Part I*,*SIAM J. Numer. Anal.*,**35:3**(1998), 990-1009. - (with
T. Manteuffel, S. McCormick, and S. Parter)
*First-order system least squares for planar linear elasticity: pure traction problem*,*SIAM J. Numer. Anal.*,**35:1**(1998), 320--335. - (with T. Manteuffel and S. McCormick)
*First-order system least squares for the Stokes equations, with application to linear elasticity*,*SIAM J. Numer. Anal.*,**34:5**(1997), 1727--1741. - (with T. Manteuffel and S. McCormick)
*First-order system least squares for second-order partial differential equations: Part II*,*SIAM J. Numer. Anal.*,**34:2**(1997), 425--454. - (with T. Manteuffel and S. McCormick)
*First-order system least squares for velocity-vorticity-pressure form of the Stokes equations, with application to linear elasticity*,*ETNA*,**3**(1995), 150--159. - (with P. Bochev, T. Manteuffel, and S. McCormick)
*First-order system least squares for the Navier-Stokes equations*, Proceedings of the Seventh Copper Moutain Conferences on Multigrid Methods, 1995, NASA Conference Publication 3339, 41-55. - (with R. Lazorov, T. Manteuffel, and S. McCormick)
*First-order system least squares for second-order partial differential equations: Part I*,*SIAM J. Numer. Anal.*,**31:6**(1994) 1785--1799.

- (with J. Yang)
*Optimal error estimate of discontinuous Galerkin methods for advection-diffusion-reaction problems with low regularity*,*Comput. Math. Appl.*, to appear. - (with S. Du)
*A finite element method for Dirichlet boundary control of elliptic partial differential equations*,*Communications in Mathematical Sciences*,**20:4**(2022) 1081-1102. - (with Z. Hao, M. Park, and G. Lin)
*Finite element method for two-sided fractional differential equations with variable coefficients: Galerkin approach*,*J. Sci. Cimput.*, 79 (2019), 700-717. - (with S. Kim and H.-C. Lee)
*Error estimate of a finite element method using stress intensity factor*,*Computers Math. Appl.*, 76:10 (2018), 2402-2408. - (with S. Zhang)
*Mixed methods for stationary Navier-Stokes equations based on pseudostress-pressure-velocity formulation*,*Math. Comp.*, 81:280 (2012), 1903-1927. - (with X. Ye and S. Zhang)
*Discontinuous Galerkin finite element methods for interface problems: a priori and a posteriori error estimations*,*SIAM J. Numer. Anal.*,**49:5**(2011), 1761-1787, CORRECTION. - (with C. Wang and S. Zhang)
*Mixed finite element methods for incompressible flow: stationary Navier-Stokes equations*,*SIAM J. Numer. Anal.*,**48:1**(2010), 79-94. - (with Y. Wang)
*Pseudostress-velocity formulation for incompressible Navier-Stokes equations*,*Int. J. Numer. Methods Fluids*,**63:3**(2010), 341-356. - (with C. Tong, P. S. Vassilevski, and C. Wang)
*Mixed finite element method for Stokes equations based on pseudostress-velocityformulation*,*Numer. Methods Part. Diff. Eqs.*,**26:4**(2010), 957-978. - (with Y. Wang)
*An error estimate for two-dimensional Stokes driven cavity problem*,*Math. Comp.*,**78:266**(2009), 771-787. - (with S. Kim, J. Pyo, and S. Kong)
*A finite element method using singular functions: interface problems*,*Hokkaido Math. J.*,**36**(2007), 815-836. *(with S. Kim, S. Kim, and S. Kong)*,*A finite element method using singular functions for Poisson equations: mixed boundary conditions*, Comput. Methods Appl. Mech. Engrg.**195:19-22**(2006), 2635-2648.- (with X. Ye)
*A mixed nonconforming finite element for linear elasticity*,*Numer. Methods Part. Diff. Eqs.*,**21:6**(2005), 1043-1051. - (with S. Kim and G. Woo)
*A finite element method using singular functions for the Poisson equation: crack singularities*,*Numer. Lin. Alg. Appl.*,**9:6-7**(2002), 445-455. - (with S. Kim)
*A finite element method using singular functions for the Poisson equation: corner singularities*,*SIAM J. Numer. Anal.*,**39:1**(2001), 286-299. - (with J. Douglas, Jr., J. E. Santos, D. Sheen, and X. Ye)
*Nonconforming quadrilateral finite elements: a correction*,*Calcolo*,**37**(2000), 253--254. - (with J. Douglas, Jr. and X. Ye)
*A stable nonconforming quadrilateral finite element method for the Stokes and Navier-Stokes equations*,*Calcolo*,**36**(1999), 215-232. *(with T. Hou and X. Wu)*(1999), 913-943.*Convergence of a multiscale finite element method for elliptic problems with rapidly oscillating coefficients*, Math. Comp.,**68:227**- (with J. Douglas, Jr.)
*Stabilized finite element methods with fast iterative solution algorithms for the Stokes problem*, Comput. Methods Appl. Mech. Engrg.,**166**(1998), 115-129.

- (with J. Douglas, Jr. and M. Park)
*Development and analysis of higher order finite volume methods over rectangles for elliptic equations*,*Advan. Comp. Math.*,**19:1-3**(2003), 3-33. - (with J. Jones, S. McCormick, and T. Russell)
*Control-volume mixed finite element methods*,*Computational Geosciences*,**1**(1997), 289-315. *On the finite volume element method*, Numer. Math.,**58**(1991), 713--735.- (with J. Mandel and S. McCormick)
*The finite volume element method for diffusion equations on general triangulations*,*SIAM J. Numer. Anal.*,**28:2**(1991), 392--402. -
*A Theoretical Foundation for the Finite Volume Element Method*, Ph.D Thesis, University of Colorado at Denver, May, 1990. - (with S. McCormick)
*On the accuracy of the finite volume element method for diffusion equations on composite grids*,*SIAM J. Numer. Anal.*,**27:3**(1990), 636--655.

- (with S. Cao and R. Falgout)
*A discretization-accurate stopping criterion for iterative solvers for finite element approximation*, submitted. - (with Y. Wang)
*A multigrid method for the pseudostress formulation of the Stokes problems*,*SIAM J. Sci. Compt.*,**29:5**(2007), 2078-2095. - (with R. Parashkevov, T. Russell, J. Wilson, and X. Ye)
*Domain decomposition for a mixed finite element method in three dimensions*,*SIAM J. Numer. Anal.*,**41:1**(2003), 181-194. - (with S. Kim and B.-C. Shin)
*Solution methods for the Poisson equation with corner singularities: numerical results*,*SIAM J. Sci. Comput.*,**23:2**(2001), 672-682. - (with R. Parashkevov, T. Russell, and X. Ye)
*Overlapping domain decomposition for a mixed finite element method in three dimensions*, In P. Bjorstad, M. Espedal, and D. Keyes (eds.)*the 9th International Conference on Domain Decomposition Methods*, Bergen, Norway, 1998, 188-196. - (with J. Mandel and S. McCormick)
*Multigrid methods for nearly singular equations and eigenvalue problems*,*SIAM J. Numer. Anal.*,**34:1**(1997), 178--200. - (with J. Douglas, Jr.)
*An analytic basis for multigrid methods for stabilized finite element methods for the Stokes problem*, in Computational Science for the 21st Century, M-O. Bristeau, G. Etgen, W. Fitzgibbon, J.L. Lions, J. P\'{e}riaux, M.F. Wheeler, Eds, Wiley, 1997, pp.~113--118. - (with G. Lai)
*Convergence analysis of multigird method for nonsymmetric and indefinite elliptic problems*,*Numer. Lin. Alg. Appl.*,**3:3**(1996), 205--220. - (with C. Goldstein and J. Pasciak)
*Multilevel iteration for mixed finite element systems with penalty*,*SIAM J. Sci. Comput.*,**14:5**(1993), 1072--1088. *Norm estimates of product operators with application to domain decomposition*,*Appl. Math. Comp.*,**53**(1993), 251--276.- (with S. McCormick)
*Schwarz alternating procedure for elliptic problems discretized by least squares mixed finite elements*, manuscript, 1993. - (with Weinan E)
*Hierarchical method for elliptic problems using wavelet*, Comm. Appl. Numer. Methods,**8**(1992), 819--825. - (with S. McCormick)
*Computational complexity of the Schwarz alternating procedure*,*Int. J. High Speed Computing*,**1**(1989), 1--28. - (with B. Y. Gao)
*Error Estimates for the Schwarz alternating procedure on L-shaped regions*, Appl. Math. Comp.,**28**(1988), 39--46. *Multigrid method with correction procedure*,*J. Comput. Math.*,**5:4**(1987), 336--341.- (with N.C. Wang)
*The revised MGE method*, Numer. Math. J. Chinese Univ.,**9:3**(1987), 227--234. - (with N.C. Wang and L.S. Kang)
*The MGE method for solving elliptic boundary value problems*, Math. Numer. Sinica,**8:1**(1986), 82--89. *Multigrid iteration with defect correction*, J. Huazhong Univ. Sci. Tech.,**14**(1986), 13--16.*Splitting extrapolation and its parallel algorithm for nonlinear elliptic boundary value problems*, Computer Sci. Engin.,**4**(1985), 1--5.*Optimal error bound for a class of comparatively generalized Hermite interpolation*, J. Engin. Math.,**2**(1985), 152--154.