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

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arxiv 2007.04542 v1 pith:UCJZGMVH submitted 2020-07-09 math.NA cs.NAcs.NE

Colby L. Wight , Jia Zhao This is my paper
classification math.NAcs.NAcs.NE
keywords pinnequationsfieldsolvingallen-cahnbeencahn-hilliardneural
verification ladder T0 review T1 audit T2 compute T3 formal
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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.

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Cited by 5 Pith papers

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

  1. Scale-Bridging Phase-Field Modeling of Microstructure Evolution by FE$^2$ Computational Homogenization

    physics.comp-ph 2026-07 conditional novelty 6.0 of 10

    A first-order FE² homogenization of phase-field theory, with micro–macro links for both the order parameter and its gradient, matches DNS evolution of averaged fields on notched and holed specimens.

  2. PIANO: Physics Informed Autoregressive Network

    cs.LG 2025-08 unverdicted novelty 5.0 of 10

    PIANO makes physics-informed neural networks autoregressive, rolling out predictions conditioned on past states under physics constraints, and claims stable, accurate long-horizon PDE and weather forecasts.

  3. Multi-Head Neural Operator for Modelling Interfacial Dynamics

    physics.comp-ph 2025-07 conditional novelty 5.0 of 10

    The Multi-Head Neural Operator predicts full phase-field trajectories in a single forward pass using time-specific projection heads with temporal connections, and outperforms FNO-2d and FNO-3d on five benchmark equations.

  4. Equivariant U-Shaped Neural Operators for the Cahn-Hilliard Phase-Field Model

    cs.LG 2025-09 conditional novelty 4.0 of 10

    E-UNO, a U-shaped Fourier neural operator with a D4 equivariance loss, predicts Cahn-Hilliard microstructure evolution more accurately than FNO and UNO baselines in reported experiments.

  5. Learning coupled Allen-Cahn and Cahn-Hilliard phase-field equations using Physics-informed neural operator(PINO)

    cs.CE 2025-07 reject novelty 4.0 of 10

    A physics-informed neural operator predicts theta-prime precipitate growth in Al-Cu alloys from three coupled phase-field equations with relative L2 errors around 1 to 6 percent on two held-out cases.

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