Cai's Publications


(I) Neural Networks in Scientific Computing (SciML)

    (0) Invited Book Chapters

  1. (with M. Liu) Physics-preserved neural network (P^2NN) method for scalar hyperbolic partial differential equations, Introduction to Scientific Machine Learning, Cambridge University Press, in progress.
  2. (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.

    3. (1) Adaptive Neural Network (ANE) Method and Approximation Theory

  3. (with J. Choi and M. Liu) ReLU neural network approximation to piecewise constant functions, arXiv:2410.16506 [math.FA].
  4. (with J. Chen and M. Liu) Self-adaptive deep neural network: numerical approximation to functions and PDEs, J. Comput. Phys., 455 (2022), 111021.
  5. (with M. Liu) Adaptive two-layer ReLU neural network: II. RITZ approximation to elliptic PDEs, Comput. Math. Appl., 113 (2022), 103-116.
  6. (with M. Liu and J. Chen) Adaptive two-layer ReLU neural network: I. Best least-squares approximation, Comput. Math. Appl., 113 (2022), 34-44.

    7. (2) Physics-Preserved Neural Network (P^2NN) Methods

    (a) Least-Squares Neural Network (LSNN) Method for Problems without Natural Minimization Principle

  7. (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., 46:4 (2024), C448-C478.
  8. (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, 21:5 (2024), 609-628.
  9. (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.
  10. (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.
  11. (with J. Chen and M. Liu) Least-squares ReLU neural network (LSNN) method for linear advection-reaction equation, J. Comput. Phys., 443 (2021), 110514.
  12. (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.

    13. (b) Deep Ritz Method and Deep Dual Methods for Problems with Natural Minimization Principle

  13. (with M. Liu and K. Ramani) Dual neural network (DuNN) method for elliptic partial differential equations and systems, J. Comput. Appl. Math., 467 (2025), 116596.
  14. (with M. Liu and K. Ramani) Deep Ritz method with adaptive quadrature for linear elasticity, Comput. Methods in Appl. Mech. Eng., 415 (2023) 116229.
  15. (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.
  16. (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.
  17. (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.

    18. (c) Evolving Neural Network (ENN) Method

  18. (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]

    19. (3) Iterative/Optimization/Training Method

  19. (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]
  20. (with A. Dokotorova, R. Falgout, and C. Herrera) Efficient shallow Ritz method for 1D diffusion problems, submitted. arXiv:2404.17750[math.NA]
  21. (with A. Dokotorova, R. Falgout, and C. Herrera) Efficient shallow Ritz method for 1D diffusion-reaction problems , submitted. arXiv:2404.01496[math.NA]
(II) Self-Adaptive Numerical Methods
  1. (with D. Cai) Hybrid a posteriori error estimator for conforming finite element approximations to stationary convection-diffusion-reaction equations, Numer. Math., 157 (2025), 477-504.
  2. (with B. Chen and J. Yang) Adaptive least-squares methods for convection-dominant diffusion-reaction problems, Comput. Math. Appl., 173 (2024), 141-150.
  3. (with S. Du) Adaptive finite element method for Dirichlet boundary control of elliptic partial differential equations, J. Sci. Comput., 89 (2021), 36.
  4. (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.
  5. (with C. He and S. Zhang) Improved ZZ error estimators for diffusion problems: discontinuous elements, Appl. Numer. Math., 159 (2021), 174-189.
  6. (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).
  7. (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.
  8. (with D. Cai) A hybrid a posteriori error estimator for conforming finite element approximations, Comput. Methods in Appl. Mech. Eng.,339 (2018), 320-340.
  9. (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.
  10. (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.
  11. (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.
  12. (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.
  13. (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.
  14. (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.
  15. (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.
  16. (with S. Zhang) Robust equilibrated residual error estimator for diffusion problems: Conforming elements, SIAM J. Numer. Anal., 50:1 (2012), 151-170.
  17. (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.
  18. (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).
  19. (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.
  20. (with S. Zhang) Recovery-based error estimators for interface problems: mixed and nonconforming finite elements, SIAM J. Numer. Anal., 48:1 (2010), 30-52.
  21. (with S. Zhang) Recovery-based error estimators for interface problems: conforming linear elements, SIAM J. Numer. Anal., 47:3 (2009), 2132-2156.
  22. (with S. Zhang) Recovery-based error estimators for interface problems: mixed and nonconforming finite elements (extended version), manuscript, 2008.
  23. (with F. Le Gland and H. Zhang) An adaptive local refinement techniques for nonlinear filtering, Publication Interne No. 954, IRISA, France, October 1995.
(III) Least-Squares Methods
  1. (with J. Ku) A dual finite element method for a singularly perturbed reaction-diffusion problem, SIAM J. Numer. Anal., 58:3 (2020) 1654-1673.
  2. (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.
  3. (with B. Chen) Least squares method for the Oseen equation, Numer. Methods Part. Diff. Eqs. , 32:4 (2016), 1289-1303.
  4. (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.
  5. (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.
  6. (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.
  7. (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.
  8. (with C. Westphal) A weighted H(div) least-squares method for second-order elliptic equations, SIAM J. Numer. Anal., 46:3 (2008), 1640-1651.
  9. (with J. Ku) The L2 norm error estimates for the Div least-squares method, SIAM J. Numer. Anal., 44:4 (2006), 1721-1734.
  10. (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.
  11. (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.
  12. (with G. Starke) Least-squares methods for linear elasticity, SIAM J. Numer. Anal., 42:2 (2004), 826-842.
  13. (with G. Starke) First-order system least squares for the stress-displacement formulation: linear elasticity, SIAM J. Numer. Anal., 41:2 (2003), 715-730.
  14. (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.
  15. (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.
  16. Least squares for the perturbed Stokes equations and the Reissner-Mindlin plate, SIAM J. Numer. Anal., 38:5 (2000), 1561--1581.
  17. (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.
  18. (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.
  19. (with X. Ye) Least-squares finite element approximations for compressible Stokes equation, Numer. Methods Part. Diff. Eqs, 16 (2000), 62-70.
  20. (with X. Ye and H. Zhang) Least-squares finite element approximations for the Reissner-Mindlin plate, Numer. Lin. Alg. Appl., 6:6 (1999), 479-496.
  21. (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.
  22. (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.
  23. (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.
  24. (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.
  25. (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.
  26. (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.
  27. (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.
(IV) Finite Element Methods
  1. (with J. Yang) Optimal error estimate of discontinuous Galerkin methods for advection-diffusion-reaction problems with low regularity , Comput. Math. Appl., 140 (2023), 282-290.
  2. (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.
  3. (with Z. Hao and Z. Zhang) Fractional-order dependent Radial basis functions meshless methods for the integral fractional Laplacian, Comput. Math. Appl., 178 (2025), 197-213.
  4. (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.
  5. (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.
  6. (with S. Zhang) Mixed methods for stationary Navier-Stokes equations based on pseudostress-pressure-velocity formulation, Math. Comp., 81:280 (2012), 1903-1927.
  7. (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.
  8. (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.
  9. (with Y. Wang) Pseudostress-velocity formulation for incompressible Navier-Stokes equations, Int. J. Numer. Methods Fluids, 63:3 (2010), 341-356.
  10. (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.
  11. (with Y. Wang) An error estimate for two-dimensional Stokes driven cavity problem, Math. Comp., 78:266 (2009), 771-787.
  12. (with S. Kim, J. Pyo, and S. Kong) A finite element method using singular functions: interface problems , Hokkaido Math. J. , 36 (2007), 815-836.
  13. (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.
  14. (with X. Ye) A mixed nonconforming finite element for linear elasticity , Numer. Methods Part. Diff. Eqs. , 21:6 (2005), 1043-1051.
  15. (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.
  16. (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.
  17. (with J. Douglas, Jr., J. E. Santos, D. Sheen, and X. Ye) Nonconforming quadrilateral finite elements: a correction, Calcolo, 37 (2000), 253--254.
  18. (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.
  19. (with T. Hou and X. Wu) Convergence of a multiscale finite element method for elliptic problems with rapidly oscillating coefficients, Math. Comp., 68:227 (1999), 913-943.
  20. (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.
(V) Finite Volume Methods
  1. (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.
  2. (with J. Jones, S. McCormick, and T. Russell) Control-volume mixed finite element methods, Computational Geosciences, 1 (1997), 289-315.
  3. On the finite volume element method, Numer. Math., 58 (1991), 713--735.
  4. (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.
  5. A Theoretical Foundation for the Finite Volume Element Method, Ph.D Thesis, University of Colorado at Denver, May, 1990.
  6. (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.
(VI) Iterative Solvers
  1. (with S. Cao and R. Falgout) A discretization-accurate stopping criterion for iterative solvers for finite element approximation , submitted.
  2. (with Y. Wang) A multigrid method for the pseudostress formulation of the Stokes problems, SIAM J. Sci. Compt., 29:5 (2007), 2078-2095.
  3. (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.
  4. (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.
  5. (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.
  6. (with J. Mandel and S. McCormick) Multigrid methods for nearly singular equations and eigenvalue problems, SIAM J. Numer. Anal., 34:1 (1997), 178--200.
  7. (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.
  8. (with G. Lai) Convergence analysis of multigird method for nonsymmetric and indefinite elliptic problems, Numer. Lin. Alg. Appl., 3:3 (1996), 205--220.
  9. (with C. Goldstein and J. Pasciak) Multilevel iteration for mixed finite element systems with penalty, SIAM J. Sci. Comput., 14:5 (1993), 1072--1088.
  10. Norm estimates of product operators with application to domain decomposition, Appl. Math. Comp., 53 (1993), 251--276.
  11. (with S. McCormick) Schwarz alternating procedure for elliptic problems discretized by least squares mixed finite elements, manuscript, 1993.
  12. (with Weinan E) Hierarchical method for elliptic problems using wavelet, Comm. Appl. Numer. Methods, 8 (1992), 819--825.
  13. (with S. McCormick) Computational complexity of the Schwarz alternating procedure, Int. J. High Speed Computing, 1 (1989), 1--28.
  14. (with B. Y. Gao) Error Estimates for the Schwarz alternating procedure on L-shaped regions, Appl. Math. Comp., 28 (1988), 39--46.
  15. Multigrid method with correction procedure, J. Comput. Math., 5:4 (1987), 336--341.
  16. (with N.C. Wang) The revised MGE method, Numer. Math. J. Chinese Univ., 9:3 (1987), 227--234.
  17. (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.
  18. Multigrid iteration with defect correction, J. Huazhong Univ. Sci. Tech., 14 (1986), 13--16.
  19. Splitting extrapolation and its parallel algorithm for nonlinear elliptic boundary value problems, Computer Sci. Engin., 4 (1985), 1--5.
  20. Optimal error bound for a class of comparatively generalized Hermite interpolation, J. Engin. Math., 2 (1985), 152--154.