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

arXiv 2007.04542 ↗First posted 9 Jul 2020math.NA cs.NA cs.NEcited by 123

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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.

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

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Cited by year (2026 to 14 Oct)

Citing papers per year: 2021: 9; 2022: 20; 2023: 26; 2024: 31; 2025: 28; 2026: 9.

123 papers.

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  1. 2022A comprehensive study of non-adaptive and residual-based adaptive sampling for physics-informed neural networkscited by 236
  2. 2022Respecting causality is all you need for training physics-informed neural networkscited by 156
  3. 2023An Expert's Guide to Training Physics-informed Neural Networkscited by 126
  4. 2021Long-time integration of parametric evolution equations with physics-informed DeepONetscited by 89
  5. 2021Multi-Objective Loss Balancing for Physics-Informed Deep Learningcited by 87
  6. 2024From PINNs to PIKANs: Recent Advances in Physics-Informed Machine Learningcited by 73
  7. 2023Residual-based attention in physics-informed neural networkscited by 72
  8. 2024PirateNets: Physics-informed Deep Learning with Residual Adaptive Networkscited by 72
  9. 2023Separable Physics-Informed Neural Networkscited by 66
  10. 2025Gradient Alignment in Physics-informed Neural Networks: A Second-Order Optimization Perspectivecited by 56
  11. 2023A unified scalable framework for causal sweeping strategies for Physics-Informed Neural Networks (PINNs) and their temporal decompositionscited by 48
  12. 2021Physics informed neural networks for continuum micromechanicscited by 46
38 more
  1. 2024ConFIG: Towards Conflict-free Training of Physics Informed Neural Networkscited by 46
  2. 2021Data-driven discovery of Green's functions with human-understandable deep learningcited by 44
  3. 2022Enhanced Physics-Informed Neural Networks with Augmented Lagrangian Relaxation Method (AL-PINNs)cited by 35
  4. 2024Finite basis Kolmogorov-Arnold networks: domain decomposition for data-driven and physics-informed problemscited by 35
  5. 2024Unveiling the optimization process of Physics Informed Neural Networks: How accurate and competitive can PINNs be?cited by 31
  6. 2021Physics-informed neural network simulation of multiphase poroelasticity using stress-split sequential trainingcited by 26
  7. 2022Lagrangian PINNs: A causality-conforming solution to failure modes of physics-informed neural networkscited by 26
  8. 2022On the Role of Fixed Points of Dynamical Systems in Training Physics-Informed Neural Networkscited by 23
  9. 2022Is L2 Physics-Informed Loss Always Suitable for Training Physics-Informed Neural Network?cited by 23
  10. 2022PIXEL: Physics-Informed Cell Representations for Fast and Accurate PDE Solverscited by 22
  11. 2023Stacked networks improve physics-informed training: applications to neural networks and deep operator networkscited by 21
  12. 2023GPT-PINN: Generative Pre-Trained Physics-Informed Neural Networks toward non-intrusive Meta-learning of parametric PDEscited by 20
  13. 2022Improved Training of Physics-Informed Neural Networks with Model Ensemblescited by 19
  14. 2022Competitive Physics Informed Networkscited by 19
  15. 2023A Framework Based on Symbolic Regression Coupled with eXtended Physics-Informed Neural Networks for Gray-Box Learning of Equations of Motion from Datacited by 18
  16. 2021Variational Physics Informed Neural Networks: the role of quadratures and test functionscited by 17
  17. 2023MultiAdam: Parameter-wise Scale-invariant Optimizer for Multiscale Training of Physics-informed Neural Networkscited by 17
  18. 2021Learning Green's Functions of Linear Reaction-Diffusion Equations with Application to Fast Numerical Solvercited by 15
  19. 2022RANG: A Residual-based Adaptive Node Generation Method for Physics-Informed Neural Networkscited by 14
  20. 2022Auto-PINN: Understanding and Optimizing Physics-Informed Neural Architecturecited by 14
  21. 2023Learning Specialized Activation Functions for Physics-informed Neural Networkscited by 14
  22. 2023Solving High Frequency and Multi-Scale PDEs with Gaussian Processescited by 14
  23. 2025Simulating Three-dimensional Turbulence with Physics-informed Neural Networkscited by 14
  24. 2022Physics-Informed Neural Networks for Shell Structurescited by 13
  25. 2023TransNet: Transferable Neural Networks for Partial Differential Equationscited by 13
  26. 2023A multifidelity approach to continual learning for physical systemscited by 13
  27. 2024Learning in PINNs: Phase transition, diffusion equilibrium, and generalizationcited by 13
  28. 2024Data-Driven Physics-Informed Neural Networks: A Digital Twin Perspectivecited by 12
  29. 2024SPIKANs: Separable Physics-Informed Kolmogorov-Arnold Networkscited by 12
  30. 2025Machine learning for modelling unstructured grid data in computational physics: a reviewcited by 12
  31. 2023Spectral operator learning for parametric PDEs without data reliancecited by 11
  32. 2023Physics-aware Machine Learning Revolutionizes Scientific Paradigm for Machine Learning and Process-based Hydrologycited by 11
  33. 2021IDRLnet: A Physics-Informed Neural Network Librarycited by 10
  34. 2022Physics-informed neural networks for solving parametric magnetostatic problemscited by 10
  35. 2023Coupling parameter and particle dynamics for adaptive sampling in Neural Galerkin schemescited by 10
  36. 2024Parametric Encoding with Attention and Convolution Mitigate Spectral Bias of Neural Partial Differential Equation Solverscited by 9
  37. 2023Feature-adjacent multi-fidelity physics-informed machine learning for partial differential equationscited by 8
  38. 2024Binary structured physics-informed neural networks for solving equations with rapidly changing solutionscited by 8

50 most cited of 123 All 123 (JSON)