A forward progressive physics informed neural network with Taylor series expansion for solving evolution partial differential equations

Wenkai Liu; Yang Liu; Wenkai Liu; Yang Liu

doi:10.3934/math.20251101

AIMS Mathematics

2025, Volume 10, Issue 10: 24857-24900. doi: 10.3934/math.20251101

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A forward progressive physics informed neural network with Taylor series expansion for solving evolution partial differential equations

Wenkai Liu ,
Yang Liu ^,

School of Mathematical Sciences, Inner Mongolia University, Hohhot, 010030, China

Received: 01 August 2025 Revised: 14 October 2025 Accepted: 24 October 2025 Published: 30 October 2025
MSC : 41A58, 65D15, 68T07

The physics informed neural network (PINN) has achieved significant success in solving evolution partial differential equations (PDEs). For improving the prediction accuracy of the PINN, we developed a new PINN with Taylor series expansion (TPINN). However, the low accuracy problem for the PINN or TPINN may occur in approximating the solution of strongly nonlinear evolution PDEs or even linear wave equations. For solving this issue, we introduced a novel efficient method, called a forward progressive PINN with Taylor series expansion (FP-TPINN), where the formula obtained by the Taylor series expansion was applied to construct extra supervised learning task and the domain decomposition in time was used to further improve the accuracy of our proposed method. We carried out several numerical experiments to demonstrate that the TPINN significantly improved the accuracy of the PINN. Moreover, we used the Korteweg-de Vries (KdV) equation to indicate that the TPINN can achieve higher accuracy than the SPINN, and illustrated that the FP-TPINN performed better than the pre-training PINN (PT-PINN) and the dimension-augmented PINN (DaPINN) by solving the Allen-Cahn equation.
- domain decomposition,
- PINN,
- Taylor series expansion,
- FP-TPINN
Citation: Wenkai Liu, Yang Liu. A forward progressive physics informed neural network with Taylor series expansion for solving evolution partial differential equations[J]. AIMS Mathematics, 2025, 10(10): 24857-24900. doi: 10.3934/math.20251101

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Abstract

The physics informed neural network (PINN) has achieved significant success in solving evolution partial differential equations (PDEs). For improving the prediction accuracy of the PINN, we developed a new PINN with Taylor series expansion (TPINN). However, the low accuracy problem for the PINN or TPINN may occur in approximating the solution of strongly nonlinear evolution PDEs or even linear wave equations. For solving this issue, we introduced a novel efficient method, called a forward progressive PINN with Taylor series expansion (FP-TPINN), where the formula obtained by the Taylor series expansion was applied to construct extra supervised learning task and the domain decomposition in time was used to further improve the accuracy of our proposed method. We carried out several numerical experiments to demonstrate that the TPINN significantly improved the accuracy of the PINN. Moreover, we used the Korteweg-de Vries (KdV) equation to indicate that the TPINN can achieve higher accuracy than the SPINN, and illustrated that the FP-TPINN performed better than the pre-training PINN (PT-PINN) and the dimension-augmented PINN (DaPINN) by solving the Allen-Cahn equation.

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