Solving Allen-Cahn and Cahn-Hilliard Equations using the Adaptive Physics Informed Neural Networks (2007.04542v1)

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.

• The paper introduces a refined PINN approach employing weighted loss functions and adaptive spatio-temporal sampling to enhance accuracy in solving phase field equations.
• The paper demonstrates that mini-batching and tailored training schemes significantly improve convergence in both one-dimensional and multi-dimensional Allen-Cahn and Cahn-Hilliard problems.
• The paper's adaptive strategies reduce computational cost and effectively manage stiffness and nonlinearity, paving the way for robust phase field simulations.

Adaptive Physics Informed Neural Networks for Phase Field Models

The paper discusses an effective approach to solving Allen-Cahn and Cahn-Hilliard equations using Adaptive Physics Informed Neural Networks (PINN). Phase field models, especially the Allen-Cahn (AC) and Cahn-Hilliard (CH) equations, are ubiquitous in modeling interfacial dynamics in various scientific realms due to their flexibility in describing interface dynamics and multiphase phenomena. Despite their generality and wide applicability, these equations often pose significant challenges in numerical approximation due to their inherent nonlinearity and stiffness caused by small parameters. This paper addresses these difficulties by proposing a refined version of PINN tailored to enhance its performance in solving phase field equations.

While PINNs have proven successful in solving various differential equations, their direct application to AC and CH equations often fails to yield accurate solutions, indicating limitations in approximation capabilities. The research advances this by integrating adaptive strategies in PINNs both spatially and temporally to enhance efficiency and precision.

Methodological Advancements

1. Weighted Loss Function: The paper introduces a loss function weighted more heavily towards initial conditions, enhancing the accuracy of early-time solutions. This approach capitalizes on the dissipative nature of phase field equations, recognizing that initial inaccuracies can propagate and amplify over subsequent time steps.
2. Mini-Batching: Incorporating mini-batch training improves convergence behavior for neural networks, leveraging stochastic gradient descent's efficacy in navigating complex loss landscapes.
3. Adaptive Sampling Strategies:
   • Spatial Adaptivity: By periodically resampling collocation points in spatial zones exhibiting high error, the PINN achieves focused learning on problematic areas, typically around sharp interface transitions.
   • Temporal Adaptivity: Two distinct temporal strategies are presented:
     • Time Sampling Approach I: This method expands the training data domain incrementally, stabilizing solutions by prioritizing earlier time intervals before progressively including later stages.
     • Time Sampling Approach II: Segregating the problem domain into discrete time intervals and training separate networks for each interval ensures localized learning, facilitating better generalization when networks are subsequently integrated.

Numerical Results

Extensive numerical experiments illustrate the superior performance of adaptive strategies in enhancing PINN efficiency and solution accuracy for AC and CH equations. Our improved PINN successfully solved both the one-dimensional and multi-dimensional benchmark problems while revealing adaptability to complex geometries. Notably, utilizing adaptive sampling strategies, our PINN overcomes the difficulties associated with sharp moving interfaces and convoluted spatial domains, achieving substantial reductions in computational costs without compromising precision.

Implications and Future Directions

This paper contributes substantially to computational methodologies for phase field modeling, offering theoretical insights into deep neural network architecture's strengths and limitations when tackling stiff partial differential equations. Looking ahead, further refinement of adaptive techniques and exploring their applicability in inverse problems are promising developments. The paper's advances cultivate a conducive environment for broadened applications of PINNs across diverse scientific fields, enabling more robust and scalable models for complex systems.

The theoretical implications of this research highlight the potential of machine learning-driven approaches to revolutionize classical numerical methods for PDEs. Practically, the improved PINN framework can enhance computational efficiency in engineering simulations, biological modeling, and materials science, fostering innovation in interdisciplinary problem solving.