Benchmark problems for physics-informed neural networks: The Allen–Cahn equation

Hyun Geun Lee; Youngjin Hwang; Yunjae Nam; Sangkwon Kim; Junseok Kim; Hyun Geun Lee; Youngjin Hwang; Yunjae Nam; Sangkwon Kim; Junseok Kim

doi:10.3934/math.2025335

AIMS Mathematics

2025, Volume 10, Issue 3: 7319-7338. doi: 10.3934/math.2025335

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Benchmark problems for physics-informed neural networks: The Allen–Cahn equation

1.
Department of Mathematics, Dongguk University, Seoul 04620, Republic of Korea
2.
Department of Mathematics, Korea University, Seoul 02841, Republic of Korea
3.
Program in Actuarial Science and Financial Engineering, Korea University, Seoul 02841, Republic of Korea

Received: 31 December 2024 Revised: 06 March 2025 Accepted: 26 March 2025 Published: 28 March 2025
MSC : 35K57, 65M22, 82C26

In this paper, we present accurate and well-designed benchmark problems for evaluating the effectiveness and precision of physics-informed neural networks (PINNs). The presented problems were generated using the Allen–Cahn (AC) equation, which models the mechanism of phase separation in binary alloy systems and simulates the temporal evolution of interfaces. The AC equation possesses the property of motion by mean curvature, which means that, in the sharp interface limit, the evolution of the interface described by the equation is governed by its mean curvature. Specifically, the velocity of the interface is proportional to its mean curvature, which implies the tendency of the interface to minimize its surface area. This property makes the AC equation a powerful mathematical model for capturing the dynamics of interface motion and phase separation processes in various physical and biological systems. The benchmark source codes for the 1D, 2D, and 3D AC equations are provided for interested researchers.
- benchmark problems,
- Allen–Cahn equation,
- physics-informed neural networks
Citation: Hyun Geun Lee, Youngjin Hwang, Yunjae Nam, Sangkwon Kim, Junseok Kim. Benchmark problems for physics-informed neural networks: The Allen–Cahn equation[J]. AIMS Mathematics, 2025, 10(3): 7319-7338. doi: 10.3934/math.2025335

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Abstract

In this paper, we present accurate and well-designed benchmark problems for evaluating the effectiveness and precision of physics-informed neural networks (PINNs). The presented problems were generated using the Allen–Cahn (AC) equation, which models the mechanism of phase separation in binary alloy systems and simulates the temporal evolution of interfaces. The AC equation possesses the property of motion by mean curvature, which means that, in the sharp interface limit, the evolution of the interface described by the equation is governed by its mean curvature. Specifically, the velocity of the interface is proportional to its mean curvature, which implies the tendency of the interface to minimize its surface area. This property makes the AC equation a powerful mathematical model for capturing the dynamics of interface motion and phase separation processes in various physical and biological systems. The benchmark source codes for the 1D, 2D, and 3D AC equations are provided for interested researchers.

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