## Publications and Reports

### 2021

An improved simple WENO limiter for discontinuous Galerkin methods solving hyperbolic systems on unstructured meshes. Jie Du, Chi-Wang Shu and Xinghui Zhong.

An Oscillation-Free Discontinuous Galerkin Method for Hyperbolic Systems. Young Liu, Jianfand Lu, and Chi-Wang Shu.

Multi-layer perceptron estimator for the total variation bounded constant in limiters for discontinuous Galerkin methods. Xinyue Yu2 and Chi-Wang Shu.

Stability Analysis of Inverse Lax-Wendroff Boundary Treatment of High Order Compact Difference Schemes for Parabolic Equations. Tingting Li, Jianfang Lu, and Chi-Wang Shu.

### 2020

Development and analysis of two new finite element schemes for a

time-domain carpet cloak model, Jichun Li and Chi-Wang Shu.

Multi-resolution HWENO schemes for hyperbolic conservation laws Jiayin Li1 , Chi-Wang Shu, and Jianxian Qiu

A high order conservative finite difference scheme for compressible two-medium flows. Feng Zheng , Chi-Wang Shu and Jianxian Qiu

Stability of high order finite difference schemes with implicit-explicit time-marching for convection-diffusion and convection-dispersion equations Meiqi Tan, Juan Cheng, and Chi-Wang Shu

Higher-order Runge-Kutta Discontinuous Galerkin methods with milti-resolution WENO limiters for solving steady-state problems. Jun Zhu, Chi-Wang Shu and Jianxian Qiu.

An Oscillation-Free Discontinuous Galerkin Method for Scalar Hyperbolic Conservation Laws. Jianfang Lu, Yong Liu, and Chi-Wang Shu.

High-order Runge-Kutta Discontinuous Galerkin methods with a new tpye of multi-resolution WENO limiters on tetrahedral meshes. Jun Zhu, Chi-Wang Shu and Jianxian Qiu

Well-posedness and H (div)-conforming finite element approximation of a linearised model for inviscid incompressible flow. G Barrenechea, E Burman, J Guzman

Convergence of Lagrange finite elements for the Maxwell Eigenvalue Problem in 2D. D Boffi, J Guzman, M Neilan

Obtaining higher-order Galerkin accuracy when the boundary is polygonally approximated. T Dupont, J Guzman, R Scott

Exact sequences on Powell–Sabin splits. J Guzmán, A Lischke, M Neilan. Calcolo 57 (2), 1-25

On the convergence and generalization of physics informed neural networks

Y Shin, J Darbon, GE Karniadakis

Fractional phase-field crystal modelling: analysis, approximation and pattern formation. IMA Journal of Applied Mathematics 85 (2), 231-262, 2020. M Ainsworth, Z Mao

On some neural network architectures that can represent viscosity solutions of certain high dimensional Hamilton--Jacobi partial differential equations. J Darbon, T Meng

On Bayesian posterior mean estimators in imaging sciences and Hamilton-Jacobi Partial Differential Equations. J Darbon, GP Langlois

On some neural network architectures that can represent viscosity solutions of certain high dimensional Hamilton--Jacobi partial differential equations. February 2020, submitted. J. Darbon, T. Meng.

"What is the fractional Laplacian?", J. Comput. Phys., Vol. 404, pp. 109009, 2020. A. Lischke, G. Pang, M. Gulian, F. Song, C. Glusa, X. Zheng, Z. Mao, W. Cai, M. M. Meerschaert, M. Ainsworth, G. E. Karniadakis

A stabilized semi-implicit Fourier spectral method for nonlinear space-fractional reaction-diffusion equations, Journal of Computational Physics 405, 10914, 2020. H Zhang, X Jiang, F Zeng, GE Karniadakis,

Extraction of mechanical properties of materials through deep learning from instrumented indentation, Proceedings of the National Academy of Sciences, March 16, 2020. L Lu, M Dao, P Kumar, U Ramamurty, GE Karniadakis, S Suresh

Physics-informed neural networks for high-speed flows, Computer Methods in Applied Mechanics and Engineering 360, 112789, 2020. Z Mao, AD Jagtap, GE Karniadakis

Adaptive activation functions accelerate convergence in deep and physics-informed neural networks, Journal of Computational Physics 404, 109136, 2020. AD Jagtap, K Kawaguchi, GE Karniadakis

Hidden fluid mechanics: Learning velocity and pressure fields from flow visualizations, Science 367 (6481), 1026-1030, 2020. M Raissi, A Yazdani, GE Karniadakis

Multiscale Modeling Meets Machine Learning: What Can We Learn?, Archives of Computational Methods in Engineering, 1-2, 2020. GCY Peng, M Alber, AB Tepole, WR Cannon, S De, GE Karniadakis, E. Kuhl

A composite neural network that learns from multi-fidelity data: Application to function approximation and inverse PDE problems, Journal of Computational Physics 401, 109020, 2020. X Meng, GE Karniadakis

Physics-Informed Generative Adversarial Networks for Stochastic Differential Equations, SIAM Journal on Scientific Computing 42 (1), A292-A317, 2020. L Yang, D Zhang, GE Karniadakis

Learning in modal space: Solving time-dependent stochastic PDEs using physics-informed neural networks, SIAM Journal on Scientific Computing 42 (2), A639-A665, 2020. D Zhang, L Guo, GE Karniadakis

An inverse Lax-Wendroff procedure for hyperbolic conservation laws with changing wind direction on the boundary. Jianfang Lu, Chi-Wang Shu, Sirui Tan, and Mengping Zhang

Weighted ghost fluid discontiuous Galerkin method for multi-medium problems. Yun-Long Liu, Chi-Wang Shu, and A-Man Zhang

Central discontinuous Galerkin methods on overlapping meshes for wave equations. Yong Liu, Jianfang Lu, Chi-Wang Shu and Mengping Zhang

A high order positivity-preserving conservative WENO remapping method on

2D quadrilateral meshes Nuo Lei, Juan Cheng and Chi-Wang Shu

High-order Runge-Kutta discontinuous Galerkin methods with a new type of multi-resolution WENO limiters on triangular meshes. Jun Zhu, Chi-Wang Shu and Jianxian Qiu

Dissipative and conservative local discontinuous Galerkin methods for the Fornberg-Whitham type equations. Qian Zhang, Yan Xu, Chi-Wang Shu

Provably Physical-Constraint-Preserving Discontinuous Galerkin Methods for Multidimensional Relativistic MHD Equations. Kailiang Wu and Chi-Wang Shu

A discontinuous Galerkin method and its error estimate for nonlinear fourth-order wave equations. Qi Tao, Yan Xu, and Chi-Wang Shu

On the conservation of finite difference WENO schemes in non-rectangular domains using the inverse Lax-Wendroff boundary treatments. Shengrong Ding, Chi-Wang Shu and Mengping Zhang

Essentially non-oscillatory and weighted essentially non-oscillatory schemes. Chi-Wang Shu

Error analysis of Runge–Kutta discontinuous Galerkin methods for linear time-dependent partial differential equations. Zheng Sun and Chi-Wang Shu

Sub-optimal convergence of discontinuous Galerkin methods with centralfluxes for even degree polynomial approximations. Yong Liu, Chai-Wang Shu and Mengping Zhang

Superconvergence Analysis of the Runge-Kutta Discontinuous Galerkin Methods for a Linear Hyperbolic Equation. Yuan Xu, Xiong Meng, Chi-Wang Shu, and Qing Zhang