Related papers: Least-Squares ReLU Neural Network (LSNN) Method Fo…
This paper studies least-squares ReLU neural network method for solving the linear advection-reaction problem with discontinuous solution. The method is a discretization of an equivalent least-squares formulation in the set of neural…
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 differential equations, specifically linear advection-reaction…
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 equivalent least-squares (LS) formulation and uses ReLU neural…
The least-squares ReLU neural network (LSNN) method was introduced and studied for solving linear advection-reaction equation with discontinuous solution in \cite{Cai2021linear,cai2023least}. The method is based on an equivalent…
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 Min Liu. ``Least-squares ReLU neural network (LSNN) method for…
In this paper we develop a non-diffusive neural network (NDNN) algorithm for accurately solving weak solutions to hyperbolic conservation laws. The principle is to construct these weak solutions by computing smooth local solutions in…
In this work, we present a subdomain discontinuous least-squares (SDLS) scheme for neutronics problems. Least-squares (LS) methods are known to be inaccurate for problems with sharp total-cross section interfaces. In addition, the…
In this paper, a least-squares finite element method for scalar nonlinear hyperbolic balance laws is proposed and studied. The approach is based on a formulation that utilizes an appropriate Helmholtz decomposition of the flux vector and is…
This article focuses on solving parametric transmission problems in one and two spatial dimensions. These problems belong to a class of partial differential equations that arise in the modeling of physical systems with heterogeneous…
Developing efficient methods for solving parametric partial differential equations is crucial for addressing inverse problems. This work introduces a Least-Squares-based Neural Network (LS-Net) method for solving linear parametric PDEs. It…
In this work, we study the problem of non-blind image deconvolution and propose a novel recurrent network architecture that leads to very competitive restoration results of high image quality. Motivated by the computational efficiency and…
In this paper, we propose a structure-guided Gauss-Newton (SgGN) method for solving least squares problems using a shallow ReLU neural network. The method effectively takes advantage of both the least squares structure and the neural…
This paper provides a least squares formulation for the training of a 2-layer convolutional neural network using quadratic activation functions, a 2-norm loss function, and no regularization term. Using this method, an analytic expression…
We propose an efficient hybrid least squares/gradient descent method to accelerate DeepONet training. Since the output of DeepONet can be viewed as linear with respect to the last layer parameters of the branch network, these parameters can…
We propose a novel model reduction approach for the approximation of non linear hyperbolic equations in the scalar and the system cases. The approach relies on an offline computation of a dictionary of solutions together with an online…
This paper proposes a novel algorithm for training recurrent neural network models of nonlinear dynamical systems from an input/output training dataset. Arbitrary convex and twice-differentiable loss functions and regularization terms are…
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 method first represents the initial data and the inflow boundary…
We propose a discontinuous least squares finite element method for solving the linear elasticity. The approximation space is obtained by patch reconstruction with only one unknown per element. We apply the L 2 norm least squares principle…
We propose an abstract discontinuous Galerkin neural network (DGNN) framework for analyzing the convergence of least-squares methods based on the residual minimization when feasible solutions are neural networks. Within this framework, we…
We propose a novel model reduction approach for the approximation of non linear hyperbolic equations in the scalar and the system cases. The approach relies on an offline computation of a dictionary of solutions together with an online…