2.5k citations · 2.9k across the 15 of their papers we have counts for
15 papers
Operator Learning Enhanced Physics-informed Neural Networks for Solving Partial Differential Equations Characterized by Sharp Solutions
Bin Lin, Zhiping Mao, Zhicheng Wang +1
Physics-informed Neural Networks (PINNs) have been shown as a promising approach for solving both forward and inverse problems of partial differential equations (PDEs). Meanwhile,…
Physics-informed neural networks with residual/gradient-based adaptive sampling methods for solving PDEs with sharp solutions
Zhiping Mao, Xuhui Meng
We consider solving the forward and inverse PDEs which have sharp solutions using physics-informed neural networks (PINNs) in this work. In particular, to better capture the sharpn…
Analysis and Hermite spectral approximation of diffusive-viscous wave equations in unbounded domains arising in geophysics
Dan Ling, Zhiping Mao
The diffusive-viscous wave equation (DVWE) is widely used in seismic exploration since it can explain frequency-dependent seismic reflections in a reservoir with hydrocarbons. Most…
Physics-informed neural networks for inverse problems in supersonic flows
Ameya D. Jagtap, Zhiping Mao, Nikolaus Adams +1
Accurate solutions to inverse supersonic compressible flow problems are often required for designing specialized aerospace vehicles. In particular, we consider the problem where we…
A comprehensive and fair comparison of two neural operators (with practical extensions) based on FAIR data
Lu Lu, Xuhui Meng, Shengze Cai +4
Neural operators can learn nonlinear mappings between function spaces and offer a new simulation paradigm for real-time prediction of complex dynamics for realistic diverse applica…
Learning Functional Priors and Posteriors from Data and Physics
Xuhui Meng, Liu Yang, Zhiping Mao +2
We develop a new Bayesian framework based on deep neural networks to be able to extrapolate in space-time using historical data and to quantify uncertainties arising from both nois…
Physics-informed neural networks (PINNs) for fluid mechanics: A review
Shengze Cai, Zhiping Mao, Zhicheng Wang +2
Despite the significant progress over the last 50 years in simulating flow problems using numerical discretization of the Navier-Stokes equations (NSE), we still cannot incorporate…
DeepM&Mnet for hypersonics: Predicting the coupled flow and finite-rate chemistry behind a normal shock using neural-network approximation of operators
Zhiping Mao, Lu Lu, Olaf Marxen +2
In high-speed flow past a normal shock, the fluid temperature rises rapidly triggering downstream chemical dissociation reactions. The chemical changes lead to appreciable changes…
DeepXDE: A deep learning library for solving differential equations
Lu Lu, Xuhui Meng, Zhiping Mao +1
Deep learning has achieved remarkable success in diverse applications; however, its use in solving partial differential equations (PDEs) has emerged only recently. Here, we present…
Nonlocal flocking dynamics: Learning the fractional order of PDEs from particle simulations
Zhiping Mao, Zhen Li, George Em Karniadakis
Flocking refers to collective behavior of a large number of interacting entities, where the interactions between discrete individuals produce collective motion on the large scale.…
Multi-domain Spectral Collocation Method for Variable-Order Nonlinear Fractional Differential Equations
Tinggang Zhao, Zhiping Mao, George Em Karniadakis
Spectral and spectral element methods using Galerkin type formulations are efficient for solving linear fractional PDEs (FPDEs) of constant order but are not efficient in solving n…
A fast solver for spectral element approximation applied to fractional differential equations using hierarchical matrix approximation
Xianjuan Li, Zhiping Mao, Fangying Song +2
We develop a fast solver for the spectral element method (SEM) applied to the two-sided fractional diffusion equation on uniform, geometric and graded meshes. By approximating the…
A spectral penalty method for two-sided fractional differential equations with general boundary conditions
Nan Wang, Zhiping Mao, Chengming Huang +1
We consider spectral approximations to the conservative form of the two-sided Riemann-Liouville (R-L) and Caputo fractional differential equations (FDEs) with nonhomogeneous Dirich…
Jacobi-Galerkin spectral method for eigenvalue problems of Riesz fractional differential equations
Lizhen Chen, Zhiping Mao, Huiyuan Li
An efficient Jacobi-Galerkin spectral method for calculating eigenvalues of Riesz fractional partial differential equations with homogeneous Dirichlet boundary values is proposed i…
What Is the Fractional Laplacian?
Anna Lischke, Guofei Pang, Mamikon Gulian +8
The fractional Laplacian in R^d has multiple equivalent characterizations. Moreover, in bounded domains, boundary conditions must be incorporated in these characterizations in math…