Solving Equations With Discontinuous Solutions Based on the AGA-PINNs Method

Liu Boyu; Jiang Linfeng; Yang Fenglian

doi:10.21656/1000-0887.460129

Volume 47 Issue 7

Jul. 2026

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Liu Boyu, Jiang Linfeng, Yang Fenglian. Solving Equations With Discontinuous Solutions Based on the AGA-PINNs Method[J]. Applied Mathematics and Mechanics, 2026, 47(7): 895-911. doi: 10.21656/1000-0887.460129

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Solving Equations With Discontinuous Solutions Based on the AGA-PINNs Method

doi: 10.21656/1000-0887.460129

School of Mathematics, Hohai University, Nanjing 210000, P.R. China

Received Date: 2025-06-25
Rev Recd Date: 2025-07-30
Publish Date: 2026-07-01

Abstract

Abstract

The physical information neural networks (PINNs) are an important tool for solving partial differential equations. In the numerical solution of partial differential equations, equations with discontinuous solutions are currently a research challenge. The PINNs and their existing improved algorithms often cannot capture the characteristics of discontinuities. However, many discontinuous problems need to be considered in the field of fluid mechanics. In response to the shortcomings of the PINNs in handling equations with discontinuous solutions, an adaptive gradient-annihilated PINNs (AGA-PINNs) method was proposed to solve the Burgers equations and the Allen-Cahn equations with discontinuous solutions. The gradient related weight functions were utilized to optimize the loss function, and the training points were dynamically adjusted based on the current residual distribution during the training process to gradually enhance the model's learning ability in discontinuous regions. The experimental results show that, the AGA-PINNs method significantly improves the accuracy of predicted physical quantities compared to the traditional PINNs, the gradient-enhanced physics-informed neural networks (gPINNs), the gradient-annihilated physics-informed neural networks (GA-PINNs) method, and the high-order deep Galerkin (DG)method. In solving the Burgers equation and the Allen-Cahn equation, the mean square errors decrease by approximately 1 order of magnitude, accurately reproducing the characteristics of shock waves in the Burgers equation and the phase separation phenomena in the Allen-Cahn equation.
- AGA-PINNs,
- Burgers equation,
- Allen-Cahn equation,
- adaptive method

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References(30)

References

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