# Energy Dissipation Preserving Physics Informed Neural Network for Allen-Cahn Equations

arXiv: https://arxiv.org/abs/2411.08760 · HTML: https://mlsanity.com/paper/2411.08760/

## Key facts

- Cited by 6 papers in the map (data through 14 Oct 2026).
- Builds on 23 of its 91 references that are in the map.
- First posted on arXiv 13 Nov 2024.

## Builds on (most cited first, 23 of 23)

- [Deep Residual Learning for Image Recognition](https://mlsanity.com/paper/1512.03385/) (2015)
- [Delving Deep into Rectifiers: Surpassing Human-Level Performance on ImageNet Classification](https://mlsanity.com/paper/1502.01852/) (2015)
- [Optuna: A Next-generation Hyperparameter Optimization Framework](https://mlsanity.com/paper/1907.10902/) (2019)
- [Optimization Methods for Large-Scale Machine Learning](https://mlsanity.com/paper/1606.04838/) (2016)
- [Automatic differentiation in machine learning: a survey](https://mlsanity.com/paper/1502.05767/) (2015)
- [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)
- [The Deep Ritz method: A deep learning-based numerical algorithm for solving variational problems](https://mlsanity.com/paper/1710.00211/) (2017)
- [Physics-informed neural networks with hard constraints for inverse design](https://mlsanity.com/paper/2102.04626/) (2021)
- [A comprehensive study of non-adaptive and residual-based adaptive sampling for physics-informed neural networks](https://mlsanity.com/paper/2207.10289/) (2022)
- [Gradient-enhanced physics-informed neural networks for forward and inverse PDE problems](https://mlsanity.com/paper/2111.02801/) (2021)
- [Why and When Can Deep -- but Not Shallow -- Networks Avoid the Curse of Dimensionality: a Review](https://mlsanity.com/paper/1611.00740/) (2016)
- [Quantifying total uncertainty in physics-informed neural networks for solving forward and inverse stochastic problems](https://mlsanity.com/paper/1809.08327/) (2018)
- [Solving Allen-Cahn and Cahn-Hilliard Equations using the Adaptive Physics Informed Neural Networks](https://mlsanity.com/paper/2007.04542/) (2020)
- [A unified deep artificial neural network approach to partial differential equations in complex geometries](https://mlsanity.com/paper/1711.06464/) (2017)
- [Estimates on the generalization error of Physics Informed Neural Networks (PINNs) for approximating a class of inverse problems for PDEs](https://mlsanity.com/paper/2007.01138/) (2020)
- [PhyCRNet: Physics-informed Convolutional-Recurrent Network for Solving Spatiotemporal PDEs](https://mlsanity.com/paper/2106.14103/) (2021)
- [A Method for Representing Periodic Functions and Enforcing Exactly Periodic Boundary Conditions with Deep Neural Networks](https://mlsanity.com/paper/2007.07442/) (2020)
- [Can Physics-Informed Neural Networks beat the Finite Element Method?](https://mlsanity.com/paper/2302.04107/) (2023)
- [Error estimates for physics informed neural networks approximating the Navier-Stokes equations](https://mlsanity.com/paper/2203.09346/) (2022)
- [Meta-learning PINN loss functions](https://mlsanity.com/paper/2107.05544/) (2021)
- [Error convergence and engineering-guided hyperparameter search of PINNs: towards optimized I-FENN performance](https://mlsanity.com/paper/2303.03918/) (2023)
- [A Priori Error Estimation of Physics-Informed Neural Networks Solving Allen--Cahn and Cahn--Hilliard Equations](https://mlsanity.com/paper/2402.02667/) (2024)

## Built on by (most cited first, 6 of 6)

- [Enforcing hidden physics in physics-informed neural networks](https://mlsanity.com/paper/2511.14348/) (2025)
- [An Operator-Consistent Graph Neural Network for Learning Diffusion Dynamics on Irregular Meshes](https://mlsanity.com/paper/2512.11860/) (2025)
- [A Structure-Preserving Framework for Solving Parabolic Partial Differential Equations with Neural Networks](https://mlsanity.com/paper/2504.10273/) (2025)
- [Physics-informed post-processing of stabilized finite element solutions for transient convection-dominated problems](https://mlsanity.com/paper/2603.03259/) (2026)
- [DeepRitzSplit Neural Operator for Phase-Field Models via Energy Splitting](https://mlsanity.com/paper/2604.18261/) (2026)
- [The limits of exactness: On the failure of automatic differentiation in physics-informed machine learning](https://mlsanity.com/paper/2609.33078/) (2026)

## Abstract (arXiv)

This paper investigates a numerical solution of Allen-Cahn equation with constant and degenerate mobility, with polynomial and logarithmic energy functionals, with deterministic and random initial functions, and with advective term in one, two, and three spatial dimensions, based on the physics-informed neural network (PINN). To improve the learning capacity of the PINN, we incorporate the energy dissipation property of the Allen-Cahn equation as a penalty term into the loss function of the network. To facilitate the learning process of random initials, we employ a continuous analogue of the initial random condition by utilizing the Fourier series expansion. Adaptive methods from traditional numerical analysis are also integrated to enhance the effectiveness of the proposed PINN. Numerical results indicate a consistent decrease in the discrete energy, while also revealing phenomena such as phase separation and metastability.