# Energy stable neural network for gradient flow equations

arXiv: https://arxiv.org/abs/2309.10002 · HTML: https://mlsanity.com/paper/2309.10002/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 11 of its 36 references that are in the map.
- First posted on arXiv 17 Sep 2023.

## Builds on (most cited first, 11 of 11)

- [Adam: A Method for Stochastic Optimization](https://mlsanity.com/paper/1412.6980/) (2014)
- [Fourier Neural Operator for Parametric Partial Differential Equations](https://mlsanity.com/paper/2010.08895/) (2020)
- [DeepONet: Learning nonlinear operators for identifying differential equations based on the universal approximation theorem of operators](https://mlsanity.com/paper/1910.03193/) (2019)
- [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)
- [Solving high-dimensional partial differential equations using deep learning](https://mlsanity.com/paper/1707.02568/) (2017)
- [MOD-Net: A Machine Learning Approach via Model-Operator-Data Network for Solving PDEs](https://mlsanity.com/paper/2107.03673/) (2021)
- [Solving Electrical Impedance Tomography with Deep Learning](https://mlsanity.com/paper/1906.03944/) (2019)
- [Friedrichs Learning: Weak Solutions of Partial Differential Equations via Deep Learning](https://mlsanity.com/paper/2012.08023/) (2020)
- [DOSnet as a Non-Black-Box PDE Solver: When Deep Learning Meets Operator Splitting](https://mlsanity.com/paper/2212.05571/) (2022)
- [Approximation of Functionals by Neural Network without Curse of Dimensionality](https://mlsanity.com/paper/2205.14421/) (2022)

## Abstract (arXiv)

We propose an energy stable network (EStable-Net) for solving gradient flow equations. The EStable-Net enables decreasing of a discrete energy along the neural network, which is consistent with the property of the gradient flow equation. The architecture of the neural network EStable-Net is based on the block network structure (Autoflow) in which output of each block can be interpreted as an intermediate state of the evolution process of the equation, and the energy stable property is incorporated in each block, which is easily generalized to include other physical and/or numerical properties. Our EStable-Net is a supervised learning network approach for solving evolution equations which does not depend on the convergence of time step goes to 0, and can be applied generally even when only data is available but the equation is unknown. We also propose a training strategy for supervised learning that employs data of the evolution stages with different nature. The EStable-Net is validated by numerical experimental results based on the Allen-Cahn equation and the Cahn-Hilliard equation in two dimensions.