# Solving Allen-Cahn and Cahn-Hilliard Equations using the Adaptive Physics Informed Neural Networks

arXiv: https://arxiv.org/abs/2007.04542 · HTML: https://mlsanity.com/paper/2007.04542/

## Key facts

- Cited by 123 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 9 Jul 2020.

## Built on by (most cited first, 30 of 123)

- [A comprehensive study of non-adaptive and residual-based adaptive sampling for physics-informed neural networks](https://mlsanity.com/paper/2207.10289/) (2022)
- [Respecting causality is all you need for training physics-informed neural networks](https://mlsanity.com/paper/2203.07404/) (2022)
- [An Expert's Guide to Training Physics-informed Neural Networks](https://mlsanity.com/paper/2308.08468/) (2023)
- [Long-time integration of parametric evolution equations with physics-informed DeepONets](https://mlsanity.com/paper/2106.05384/) (2021)
- [Multi-Objective Loss Balancing for Physics-Informed Deep Learning](https://mlsanity.com/paper/2110.09813/) (2021)
- [From PINNs to PIKANs: Recent Advances in Physics-Informed Machine Learning](https://mlsanity.com/paper/2410.13228/) (2024)
- [Residual-based attention in physics-informed neural networks](https://mlsanity.com/paper/2307.00379/) (2023)
- [PirateNets: Physics-informed Deep Learning with Residual Adaptive Networks](https://mlsanity.com/paper/2402.00326/) (2024)
- [Separable Physics-Informed Neural Networks](https://mlsanity.com/paper/2306.15969/) (2023)
- [Gradient Alignment in Physics-informed Neural Networks: A Second-Order Optimization Perspective](https://mlsanity.com/paper/2502.00604/) (2025)
- [A unified scalable framework for causal sweeping strategies for Physics-Informed Neural Networks (PINNs) and their temporal decompositions](https://mlsanity.com/paper/2302.14227/) (2023)
- [Physics informed neural networks for continuum micromechanics](https://mlsanity.com/paper/2110.07374/) (2021)
- [ConFIG: Towards Conflict-free Training of Physics Informed Neural Networks](https://mlsanity.com/paper/2408.11104/) (2024)
- [Data-driven discovery of Green's functions with human-understandable deep learning](https://mlsanity.com/paper/2105.00266/) (2021)
- [Enhanced Physics-Informed Neural Networks with Augmented Lagrangian Relaxation Method (AL-PINNs)](https://mlsanity.com/paper/2205.01059/) (2022)
- [Finite basis Kolmogorov-Arnold networks: domain decomposition for data-driven and physics-informed problems](https://mlsanity.com/paper/2406.19662/) (2024)
- [Unveiling the optimization process of Physics Informed Neural Networks: How accurate and competitive can PINNs be?](https://mlsanity.com/paper/2405.04230/) (2024)
- [Physics-informed neural network simulation of multiphase poroelasticity using stress-split sequential training](https://mlsanity.com/paper/2110.03049/) (2021)
- [Lagrangian PINNs: A causality-conforming solution to failure modes of physics-informed neural networks](https://mlsanity.com/paper/2205.02902/) (2022)
- [On the Role of Fixed Points of Dynamical Systems in Training Physics-Informed Neural Networks](https://mlsanity.com/paper/2203.13648/) (2022)
- [Is L2 Physics-Informed Loss Always Suitable for Training Physics-Informed Neural Network?](https://mlsanity.com/paper/2206.02016/) (2022)
- [PIXEL: Physics-Informed Cell Representations for Fast and Accurate PDE Solvers](https://mlsanity.com/paper/2207.12800/) (2022)
- [Stacked networks improve physics-informed training: applications to neural networks and deep operator networks](https://mlsanity.com/paper/2311.06483/) (2023)
- [GPT-PINN: Generative Pre-Trained Physics-Informed Neural Networks toward non-intrusive Meta-learning of parametric PDEs](https://mlsanity.com/paper/2303.14878/) (2023)
- [Improved Training of Physics-Informed Neural Networks with Model Ensembles](https://mlsanity.com/paper/2204.05108/) (2022)
- [Competitive Physics Informed Networks](https://mlsanity.com/paper/2204.11144/) (2022)
- [A Framework Based on Symbolic Regression Coupled with eXtended Physics-Informed Neural Networks for Gray-Box Learning of Equations of Motion from Data](https://mlsanity.com/paper/2305.10706/) (2023)
- [Variational Physics Informed Neural Networks: the role of quadratures and test functions](https://mlsanity.com/paper/2109.02035/) (2021)
- [MultiAdam: Parameter-wise Scale-invariant Optimizer for Multiscale Training of Physics-informed Neural Networks](https://mlsanity.com/paper/2306.02816/) (2023)
- [Learning Green's Functions of Linear Reaction-Diffusion Equations with Application to Fast Numerical Solver](https://mlsanity.com/paper/2105.11045/) (2021)

## Abstract (arXiv)

Phase field models, in particular, the Allen-Cahn type and Cahn-Hilliard type equations, have been widely used to investigate interfacial dynamic problems. Designing accurate, efficient, and stable numerical algorithms for solving the phase field models has been an active field for decades. In this paper, we focus on using the deep neural network to design an automatic numerical solver for the Allen-Cahn and Cahn-Hilliard equations by proposing an improved physics informed neural network (PINN). Though the PINN has been embraced to investigate many differential equation problems, we find a direct application of the PINN in solving phase-field equations won't provide accurate solutions in many cases. Thus, we propose various techniques that add to the approximation power of the PINN. As a major contribution of this paper, we propose to embrace the adaptive idea in both space and time and introduce various sampling strategies, such that we are able to improve the efficiency and accuracy of the PINN on solving phase field equations. In addition, the improved PINN has no restriction on the explicit form of the PDEs, making it applicable to a wider class of PDE problems, and shedding light on numerical approximations of other PDEs in general.