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

2020/07/09 by Colby Wight, Colby L. Wight, Jia Zhao +2 · 34 citations
Computer Science · Materials Science · Mathematics · Physics and Astronomy · #FOS: Computer and information sciences #FOS: Mathematics #Magnetic Properties and Applications #Model Reduction and Neural Networks #Neural and Evolutionary Computing (cs.NE) #Numerical Analysis (math.NA) #Solidification and crystal growth phenomena #cs.NA #cs.NE #math.NA

paper · pdf · doi:10.48550/arxiv.2007.04542

arxiv created 2020/07/09 · openalex publication_date 2020/07/09 · arxiv updated 2020/07/10 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

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.

Citations

• Multistep Neural Networks for Data-driven Discovery of Nonlinear\n Dynamical Systems
• Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations
• Activation Functions: Comparison of trends in Practice and Research for Deep Learning

Cited by

• Physics-Informed Neural Networks for Joint Source and Parameter Estimation in Advection-Diffusion Equations
• DAE-HardNet: A Physics Constrained Neural Network Enforcing Differential-Algebraic Hard Constraints
• A Multi-Phase Dual-PINN Framework: Soft Boundary-Interior Specialization via Distance-Weighted Priors
• Discontinuity-aware physics-informed neural network for phase-field method in three-phase flow with phase change
• A Global Spacetime Optimization Approach to the Real-Space Time-Dependent Schrödinger Equation
• Enforcing hidden physics in physics-informed neural networks
• Self-adaptive weighting and sampling for physics-informed neural networks
• Auto-Adaptive PINNs with Applications to Phase Transitions
• A DeepLagrangian method for learning and generating aggregation patterns in multi-dimensional Keller-Segel chemotaxis systems
• AB-PINNs: Adaptive-Basis Physics-Informed Neural Networks for Residual-Driven Domain Decomposition
• Solving Fokker-Planck-Kolmogorov Equation by Distribution Self-adaptation Normalized Physics-informed Neural Networks
• Parallel-in-Time Solution of Allen-Cahn Equations by Integrating Operator Learning into the Parareal Method
• Multi-Objective Loss Balancing in Physics-Informed Neural Networks for Fluid Flow Applications
• PIANO: Physics Informed Autoregressive Network
• Equivariant U-Shaped Neural Operators for the Cahn-Hilliard Phase-Field Model
• SSBE-PINN: A Sobolev Boundary Scheme Boosting Stability and Accuracy in Elliptic/Parabolic PDE Learning
• Intelligent Sampling of Extreme-Scale Turbulence Datasets for Accurate and Efficient Spatiotemporal Model Training
• Learning coupled Allen-Cahn and Cahn-Hilliard phase-field equations using Physics-informed neural operator(PINO)
• Improving physics-informed neural network extrapolation via transfer learning and adaptive activation functions
• A Residual Guided strategy with Generative Adversarial Networks in training Physics-Informed Transformer Networks
• Energy Dissipation Rate Guided Adaptive Sampling for Physics-Informed Neural Networks: Resolving Surface-Bulk Dynamics in Allen-Cahn Systems
• Simulating Three-dimensional Turbulence with Physics-informed Neural Networks
• Over-PINNs: Enhancing Physics-Informed Neural Networks via Higher-Order Partial Derivative Overdetermination of PDEs
• Physics-informed machine learning
• A comprehensive analysis of PINNs: Variants, Applications, and Challenges
• Scientific machine learning in Hydrology: a unified perspective
• Convergence Guarantees for Gradient-Based Training of Neural PDE Solvers: From Linear to Nonlinear PDEs
• Convergence Properties of PINNs for the Navier-Stokes-Cahn-Hilliard System
• Learning and Transferring Physical Models through Derivatives
• Di-BiLPS: Denoising induced Bidirectional Latent-PDE-Solver under Sparse Observations
• Reliable and efficient inverse analysis using physics-informed neural networks with normalized distance functions and adaptive weight tuning
• PINNs for Solving Unsteady Maxwell's Equations: Convergence Issues and Comparative Assessment with Compact Schemes
• A Structure-Preserving Framework for Solving Parabolic Partial Differential Equations with Neural Networks
• BO-SA-PINNs: Self-adaptive physics-informed neural networks based on Bayesian optimization for automatically designing PDE solvers

Related

• Numerical Approximations of the Allen-Cahn-Ohta-Kawasaki (ACOK) Equation with Modified Physics Informed Neural Networks (PINNs)
• A Priori Error Estimation of Physics-Informed Neural Networks Solving Allen--Cahn and Cahn--Hilliard Equations
• Learning coupled Allen-Cahn and Cahn-Hilliard phase-field equations using Physics-informed neural operator(PINO)
• Mass-preserving spatio-temporal adaptive PINN for Cahn-Hilliard equations with strong nonlinearity and singularity
• An End-to-End Deep Learning Method for Solving Nonlocal Allen-Cahn and Cahn-Hilliard Phase-Field Models