20 citations · 41 across the 11 of their papers we have counts for
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Least-Squares Neural Network (LSNN) Method for Scalar Hyperbolic Partial Differential Equations
Min Liu, Zhiqiang Cai
This chapter offers a comprehensive introduction to the least-squares neural network (LSNN) method introduced in [14,16], for solving scalar first-order hyperbolic partial differen…
Evolving Neural Network (ENN) Method for One-Dimensional Scalar Hyperbolic Conservation Laws: I Linear and Quadratic Fluxes
Zhiqiang Cai, Brooke Hejnal
We propose and study the evolving neural network (ENN) method for solving one-dimensional scalar hyperbolic conservation laws with linear and quadratic spatial fluxes. The ENN meth…
Least-Squares Neural Network (LSNN) Method For Linear Advection-Reaction Equation: Non-constant Jumps
Zhiqiang Cai, Junpyo Choi, Min Liu
The least-squares ReLU neural network (LSNN) method was introduced and studied for solving linear advection-reaction equation with discontinuous solution in \cite{Cai2021linear,cai…
Deep Ritz Method with Adaptive Quadrature for Linear Elasticity
Min Liu, Zhiqiang Cai, Karthik Ramani
In this paper, we study the deep Ritz method for solving the linear elasticity equation from a numerical analysis perspective. A modified Ritz formulation using the …
Least-Squares Neural Network (LSNN) Method For Linear Advection-Reaction Equation: Discontinuity Interface
Zhiqiang Cai, Junpyo Choi, Min Liu
We studied the least-squares ReLU neural network (LSNN) method for solving linear advection-reaction equation with discontinuous solution in [Cai, Zhiqiang, Jingshuang Chen, and Mi…
Least-Squares Neural Network (LSNN) Method For Scalar Nonlinear Hyperbolic Conservation Laws: Discrete Divergence Operator
Zhiqiang Cai, Jingshuang Chen, Min Liu
A least-squares neural network (LSNN) method was introduced for solving scalar linear and nonlinear hyperbolic conservation laws (HCLs) in [7, 6]. This method is based on an equiva…
Self-adaptive deep neural network: Numerical approximation to functions and PDEs
Zhiqiang Cai, Jingshuang Chen, Min Liu
Designing an optimal deep neural network for a given task is important and challenging in many machine learning applications. To address this issue, we introduce a self-adaptive al…