# Cell-average based neural network method for hyperbolic and parabolic partial differential equations

arXiv: https://arxiv.org/abs/2107.00813 · HTML: https://mlsanity.com/paper/2107.00813/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Builds on 20 of its 50 references that are in the map.
- First posted on arXiv 2 Jul 2021.

## Builds on (most cited first, 20 of 20)

- [DGM: A deep learning algorithm for solving partial differential equations](https://mlsanity.com/paper/1708.07469/) (2017)
- [DeepXDE: A deep learning library for solving differential equations](https://mlsanity.com/paper/1907.04502/) (2019)
- [PDE-Net 2.0: Learning PDEs from Data with A Numeric-Symbolic Hybrid Deep Network](https://mlsanity.com/paper/1812.04426/) (2018)
- [Deep Neural Networks Motivated by Partial Differential Equations](https://mlsanity.com/paper/1804.04272/) (2018)
- [A unified deep artificial neural network approach to partial differential equations in complex geometries](https://mlsanity.com/paper/1711.06464/) (2017)
- [Learning Neural PDE Solvers with Convergence Guarantees](https://mlsanity.com/paper/1906.01200/) (2019)
- [A Theoretical Analysis of Deep Neural Networks and Parametric PDEs](https://mlsanity.com/paper/1904.00377/) (2019)
- [Reversible Architectures for Arbitrarily Deep Residual Neural Networks](https://mlsanity.com/paper/1709.03698/) (2017)
- [Physics Informed Extreme Learning Machine (PIELM) -- A rapid method for the numerical solution of partial differential equations](https://mlsanity.com/paper/1907.03507/) (2019)
- [Machine learning approximation algorithms for high-dimensional fully nonlinear partial differential equations and second-order backward stochastic differential equations](https://mlsanity.com/paper/1709.05963/) (2017)
- [Data-Driven Deep Learning of Partial Differential Equations in Modal Space](https://mlsanity.com/paper/1910.06948/) (2019)
- [Deep learning observables in computational fluid dynamics](https://mlsanity.com/paper/1903.03040/) (2019)
- [Machine Learning for semi linear PDEs](https://mlsanity.com/paper/1809.07609/) (2018)
- [Deep Relaxation: partial differential equations for optimizing deep neural networks](https://mlsanity.com/paper/1704.04932/) (2017)
- [Neural networks-based backward scheme for fully nonlinear PDEs](https://mlsanity.com/paper/1908.00412/) (2019)
- [Least-Squares ReLU Neural Network (LSNN) Method For Linear Advection-Reaction Equation](https://mlsanity.com/paper/2105.11632/) (2021)
- [Learning to Discretize: Solving 1D Scalar Conservation Laws via Deep Reinforcement Learning](https://mlsanity.com/paper/1905.11079/) (2019)
- [Variational training of neural network approximations of solution maps for physical models](https://mlsanity.com/paper/1905.02789/) (2019)
- [A Neural Network based Shock Detection and Localization Approach for Discontinuous Galerkin Methods](https://mlsanity.com/paper/2001.08201/) (2020)
- [Least-Squares ReLU Neural Network (LSNN) Method For Scalar Nonlinear Hyperbolic Conservation Law](https://mlsanity.com/paper/2105.11627/) (2021)

## Built on by (most cited first, 1 of 1)

- [Learning Domain-Independent Green's Function For Elliptic Partial Differential Equations](https://mlsanity.com/paper/2401.17172/) (2024)

## Abstract (arXiv)

Motivated by finite volume scheme, a cell-average based neural network method is proposed. The method is based on the integral or weak formulation of partial differential equations. A simple feed forward network is forced to learn the solution average evolution between two neighboring time steps. Offline supervised training is carried out to obtain the optimal network parameter set, which uniquely identifies one finite volume like neural network method. Once well trained, the network method is implemented as a finite volume scheme, thus is mesh dependent. Different to traditional numerical methods, our method can be relieved from the explicit scheme CFL restriction and can adapt to any time step size for solution evolution. For Heat equation, first order of convergence is observed and the errors are related to the spatial mesh size but are observed independent of the mesh size in time. The cell-average based neural network method can sharply evolve contact discontinuity with almost zero numerical diffusion introduced. Shock and rarefaction waves are well captured for nonlinear hyperbolic conservation laws.