# Least-Squares ReLU Neural Network (LSNN) Method For Scalar Nonlinear Hyperbolic Conservation Law

arXiv: https://arxiv.org/abs/2105.11627 · HTML: https://mlsanity.com/paper/2105.11627/

## Key facts

- Cited by 7 papers in the map (data through 14 Oct 2026).
- Builds on 6 of its 41 references that are in the map.
- First posted on arXiv 25 May 2021.

## Builds on (most cited first, 6 of 6)

- [Adam: A Method for Stochastic Optimization](https://mlsanity.com/paper/1412.6980/) (2014)
- [DGM: A deep learning algorithm for solving partial differential equations](https://mlsanity.com/paper/1708.07469/) (2017)
- [The Deep Ritz method: A deep learning-based numerical algorithm for solving variational problems](https://mlsanity.com/paper/1710.00211/) (2017)
- [Physics Informed Deep Learning (Part I): Data-driven Solutions of Nonlinear Partial Differential Equations](https://mlsanity.com/paper/1711.10561/) (2017)
- [Deep least-squares methods: an unsupervised learning-based numerical method for solving elliptic PDEs](https://mlsanity.com/paper/1911.02109/) (2019)
- [Learning to Discretize: Solving 1D Scalar Conservation Laws via Deep Reinforcement Learning](https://mlsanity.com/paper/1905.11079/) (2019)

## Built on by (most cited first, 7 of 7)

- [Physics-Guided, Physics-Informed, and Physics-Encoded Neural Networks in Scientific Computing](https://mlsanity.com/paper/2211.07377/) (2022)
- [Physics consistent machine learning framework for inverse modeling with applications to ICF capsule implosions](https://mlsanity.com/paper/2412.20192/) (2024)
- [A Structure-Guided Gauss-Newton Method for Shallow ReLU Neural Network](https://mlsanity.com/paper/2404.05064/) (2024)
- [Self-adaptive deep neural network: Numerical approximation to functions and PDEs](https://mlsanity.com/paper/2109.02839/) (2021)
- [Least-Squares Neural Network (LSNN) Method For Scalar Nonlinear Hyperbolic Conservation Laws: Discrete Divergence Operator](https://mlsanity.com/paper/2110.10895/) (2021)
- [Cell-average based neural network method for hyperbolic and parabolic partial differential equations](https://mlsanity.com/paper/2107.00813/) (2021)
- [Efficient Weak-Entropy PINN for Solving Hyperbolic Conservation Laws](https://mlsanity.com/paper/2608.10389/) (2026)

## Abstract (arXiv)

We introduced the least-squares ReLU neural network (LSNN) method for solving the linear advection-reaction problem with discontinuous solution and showed that the method outperforms mesh-based numerical methods in terms of the number of degrees of freedom. This paper studies the LSNN method for scalar nonlinear hyperbolic conservation law. The method is a discretization of an equivalent least-squares (LS) formulation in the set of neural network functions with the ReLU activation function. Evaluation of the LS functional is done by using numerical integration and conservative finite volume scheme. Numerical results of some test problems show that the method is capable of approximating the discontinuous interface of the underlying problem automatically through the free breaking lines of the ReLU neural network. Moreover, the method does not exhibit the common Gibbs phenomena along the discontinuous interface.