mlsanityData to 14 Oct 2026

Energy Dissipation Preserving Physics Informed Neural Network for Allen-Cahn Equations

arXiv 2411.08760 ↗First posted 13 Nov 2024math.NA cs.LG cs.NAcited by 6

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Abstract (arXiv)

This paper investigates a numerical solution of Allen-Cahn equation with constant and degenerate mobility, with polynomial and logarithmic energy functionals, with deterministic and random initial functions, and with advective term in one, two, and three spatial dimensions, based on the physics-informed neural network (PINN). To improve the learning capacity of the PINN, we incorporate the energy dissipation property of the Allen-Cahn equation as a penalty term into the loss function of the network. To facilitate the learning process of random initials, we employ a continuous analogue of the initial random condition by utilizing the Fourier series expansion. Adaptive methods from traditional numerical analysis are also integrated to enhance the effectiveness of the proposed PINN. Numerical results indicate a consistent decrease in the discrete energy, while also revealing phenomena such as phase separation and metastability.

Key facts. Cited by 6 papers in the map (data through 14 Oct 2026). Builds on 23 of its 91 references that are in the map. First posted on arXiv 13 Nov 2024.

Builds on

23 of 91 references.

References · 23

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11 more
  1. 2018Quantifying total uncertainty in physics-informed neural networks for solving forward and inverse stochastic problemscited by 147
  2. 2020Solving Allen-Cahn and Cahn-Hilliard Equations using the Adaptive Physics Informed Neural Networkscited by 123
  3. 2017A unified deep artificial neural network approach to partial differential equations in complex geometriescited by 122
  4. 2020Estimates on the generalization error of Physics Informed Neural Networks (PINNs) for approximating a class of inverse problems for PDEscited by 102
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  9. 2021Meta-learning PINN loss functionscited by 51
  10. 2023Error convergence and engineering-guided hyperparameter search of PINNs: towards optimized I-FENN performancecited by 5
  11. 2024A Priori Error Estimation of Physics-Informed Neural Networks Solving Allen--Cahn and Cahn--Hilliard Equationscited by 1

Built on by

Cited by year (2026 to 14 Oct)

Citing papers per year: 2025: 3; 2026: 3.

6 papers.

  1. 2025Enforcing hidden physics in physics-informed neural networkscited by 3
  2. 2025An Operator-Consistent Graph Neural Network for Learning Diffusion Dynamics on Irregular Meshescited by 1
  3. 2025A Structure-Preserving Framework for Solving Parabolic Partial Differential Equations with Neural Networkscited by 0
  4. 2026Physics-informed post-processing of stabilized finite element solutions for transient convection-dominated problemscited by 0
  5. 2026DeepRitzSplit Neural Operator for Phase-Field Models via Energy Splittingcited by 0
  6. 2026The limits of exactness: On the failure of automatic differentiation in physics-informed machine learningcited by 0