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<p>Compared to multi-layer neural networks based on nonlinear activation functions, single-layer neural networks that use smooth linear functions as activation functions can quickly numerically solve time-fractional partial differential equations (TFPDEs) with time derivative of order <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\(\alpha \in (0,1)\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>α</mi> <mo>∈</mo> <mo stretchy="false">(</mo> <mn>0</mn> <mo>,</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation>. However, due to the weak singularity of the solution which is not smooth near the initial time, resulting in significant numerical errors when using neural networks based on smooth functions as activation functions. In this paper, we develop a decomposition method to reduce the singularity of the initial value, and theoretically demonstrate that this method can reduce the singularity. Then, a new single-layer neural network and a fast training method are proposed based on the decomposition method and the smooth linear activation function. Numerical experiments have shown that this method reduces errors and improves computational speed.</p>

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Efficient Legendre Polynomial Neural Networks Method for Time-Fractional Partial Differential Equations with Singularity

  • 期刊论文

  • 发表日期：

    2025年9月13日

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  • Hong-Liang Huang,
  • Da-Kang Cen,
  • Seak-Weng Vong,
  • Siu-Long Lei

摘要

Compared to multi-layer neural networks based on nonlinear activation functions, single-layer neural networks that use smooth linear functions as activation functions can quickly numerically solve time-fractional partial differential equations (TFPDEs) with time derivative of order \(\alpha \in (0,1)\) α ∈ ( 0 , 1 ) . However, due to the weak singularity of the solution which is not smooth near the initial time, resulting in significant numerical errors when using neural networks based on smooth functions as activation functions. In this paper, we develop a decomposition method to reduce the singularity of the initial value, and theoretically demonstrate that this method can reduce the singularity. Then, a new single-layer neural network and a fast training method are proposed based on the decomposition method and the smooth linear activation function. Numerical experiments have shown that this method reduces errors and improves computational speed.