# Invariant deep neural networks under the finite group for solving partial differential equations

arXiv: https://arxiv.org/abs/2407.20560 · HTML: https://mlsanity.com/paper/2407.20560/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 13 of its 46 references that are in the map.
- First posted on arXiv 30 Jul 2024.

## Builds on (most cited first, 13 of 13)

- [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)
- [When and why PINNs fail to train: A neural tangent kernel perspective](https://mlsanity.com/paper/2007.14527/) (2020)
- [DeepXDE: A deep learning library for solving differential equations](https://mlsanity.com/paper/1907.04502/) (2019)
- [The Deep Ritz method: A deep learning-based numerical algorithm for solving variational problems](https://mlsanity.com/paper/1710.00211/) (2017)
- [hp-VPINNs: Variational Physics-Informed Neural Networks With Domain Decomposition](https://mlsanity.com/paper/2003.05385/) (2020)
- [Gradient-enhanced physics-informed neural networks for forward and inverse PDE problems](https://mlsanity.com/paper/2111.02801/) (2021)
- [Understanding and mitigating gradient pathologies in physics-informed neural networks](https://mlsanity.com/paper/2001.04536/) (2020)
- [Universal approximations of invariant maps by neural networks](https://mlsanity.com/paper/1804.10306/) (2018)
- [Parallel Physics-Informed Neural Networks via Domain Decomposition](https://mlsanity.com/paper/2104.10013/) (2021)
- [DPM: A Novel Training Method for Physics-Informed Neural Networks in Extrapolation](https://mlsanity.com/paper/2012.02681/) (2020)
- [Enforcing continuous symmetries in physics-informed neural network for solving forward and inverse problems of partial differential equations](https://mlsanity.com/paper/2206.09299/) (2022)
- [JAX-DIPS: Neural bootstrapping of finite discretization methods and application to elliptic problems with discontinuities](https://mlsanity.com/paper/2210.14312/) (2022)

## Abstract (arXiv)

Utilizing physics-informed neural networks (PINN) to solve partial differential equations (PDEs) becomes a hot issue and also shows its great powers, but still suffers from the dilemmas of limited predicted accuracy in the sampling domain and poor prediction ability beyond the sampling domain which are usually mitigated by adding the physical properties of PDEs into the loss function or by employing smart techniques to change the form of loss function for special PDEs. In this paper, we design a symmetry-enhanced deep neural network (sDNN) which makes the architecture of neural networks invariant under the finite group through expanding the dimensions of weight matrixes and bias vectors in each hidden layers by the order of finite group if the group has matrix representations, otherwise extending the set of input data and the hidden layers except for the first hidden layer by the order of finite group. However, the total number of training parameters is only about one over the order of finite group of the original PINN size due to the symmetric architecture of sDNN. Furthermore, we give special forms of weight matrixes and bias vectors of sDNN, and rigorously prove that the architecture itself is invariant under the finite group and the sDNN has the universal approximation ability to learn the function keeping the finite group. Numerical results show that the sDNN has strong predicted abilities in and beyond the sampling domain and performs far better than the vanilla PINN with fewer training points and simpler architecture.