mlsanityData to 14 Oct 2026

Least-Squares ReLU Neural Network (LSNN) Method For Linear Advection-Reaction Equation

arXiv 2105.11632 ↗First posted 25 May 2021math.NA cs.LG cs.NAcited by 16

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Abstract (arXiv)

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 network functions with the ReLU activation function. The method is capable of approximating the discontinuous interface of the underlying problem automatically through the free hyper-planes of the ReLU neural network and, hence, outperforms mesh-based numerical methods in terms of the number of degrees of freedom. Numerical results of some benchmark test problems show that the method can not only approximate the solution with the least number of parameters, but also avoid the common Gibbs phenomena along the discontinuous interface. Moreover, a three-layer ReLU neural network is necessary and sufficient in order to well approximate a discontinuous solution with an interface in ℝ2 that is not a straight line.

Key facts. Cited by 16 papers in the map (data through 14 Oct 2026). Builds on 4 of its 30 references that are in the map. First posted on arXiv 25 May 2021.

Builds on

4 of 30 references.

References · 4

  1. 2014Adam: A Method for Stochastic Optimizationcited by 37,240
  2. 2017DGM: A deep learning algorithm for solving partial differential equationscited by 678
  3. 2019Deep Network Approximation Characterized by Number of Neuronscited by 122
  4. 2019Deep least-squares methods: an unsupervised learning-based numerical method for solving elliptic PDEscited by 28

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Citing papers per year: 2021: 4; 2022: 3; 2023: 2; 2024: 6; 2025: 1.

16 papers.

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  1. 2022Bounding the Rademacher Complexity of Fourier neural operatorscited by 16
  2. 2024Approximating Numerical Fluxes Using Fourier Neural Operators for Hyperbolic Conservation Lawscited by 11
  3. 2021DeepParticle: learning invariant measure by a deep neural network minimizing Wasserstein distance on data generated from an interacting particle methodcited by 9
  4. 2022Solving Elliptic Problems with Singular Sources using Singularity Splitting Deep Ritz Methodcited by 4
  5. 2023Capturing the Diffusive Behavior of the Multiscale Linear Transport Equations by Asymptotic-Preserving Convolutional DeepONetscited by 3
  6. 2023Solving Elliptic Optimal Control Problems via Neural Networks and Optimality Systemcited by 3
  7. 2024A Structure-Guided Gauss-Newton Method for Shallow ReLU Neural Networkcited by 3
  8. 2021Self-adaptive deep neural network: Numerical approximation to functions and PDEscited by 2
  9. 2021Least-Squares Neural Network (LSNN) Method For Scalar Nonlinear Hyperbolic Conservation Laws: Discrete Divergence Operatorcited by 2
  10. 2022A variational neural network approach for glacier modelling with nonlinear rheologycited by 2
  11. 2021Cell-average based neural network method for hyperbolic and parabolic partial differential equationscited by 1
  12. 2024Neural Networks Trained by Weight Permutation are Universal Approximatorscited by 1
4 more
  1. 2025An Iterative Deep Ritz Method for Monotone Elliptic Problemscited by 1
  2. 2024Macroscopic auxiliary asymptotic preserving neural networks for the linear radiative transfer equationscited by 0
  3. 2024Efficient Shallow Ritz Method For 1D Diffusion-Reaction Problemscited by 0
  4. 2024ReLU neural network approximation to piecewise constant functionscited by 0