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<p>A&#xa0;nonlinear Klein-Gordon equation is used in many science and engineering issues. In this inquiry, we propose a&#xa0;deep learning method, i.e., multi-term physics-informed neural networks (PINNs), to resolve forward and inverse problems of multi-dimensional hyperbolic nonlinear Klein-Gordon equations. The PINNs algorithm incorporates the residuals of the issue, initial circumstances, and boundary conditions pointing towards the training process. The proposed scheme has been trained with a&#xa0;view of minimizing the total loss due to multi-term loss functional by a&#xa0;series of densely connected neural networks which are also referred to as feed-forward deep neural networks. To illustrate the effectiveness and usages of our proposed approach, four computational examples are given. We compared the proposed scheme with the analytical or true solution and other methods existing in the literature. The findings indicate that the suggested approach provides more accurate results and is an efficient deep learning method, offering a&#xa0;reliable and accurate estimation for both forward and inverse problems of multi-dimensional non-linear Klein-Gordon equations. Therefore, the proposed computational approach is efficient in solving challenging physical forward and inverse problems with different boundary and initial conditions.</p>

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Leveraging Advanced Deep Neural Networks Algorithm for Solving Multi-dimensional Forward and Inverse Problems of Klein-Gordon Equation with Quadratic, Cubic, and Fifth-Degree Polynomials Nonlinearity

  • 期刊论文

  • 发表日期：

    2025年9月1日

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  • Alemayehu Tamirie Deresse,
  • Alemu Senbeta Bekela,
  • Tamirat Temesgen Dufera

摘要

A nonlinear Klein-Gordon equation is used in many science and engineering issues. In this inquiry, we propose a deep learning method, i.e., multi-term physics-informed neural networks (PINNs), to resolve forward and inverse problems of multi-dimensional hyperbolic nonlinear Klein-Gordon equations. The PINNs algorithm incorporates the residuals of the issue, initial circumstances, and boundary conditions pointing towards the training process. The proposed scheme has been trained with a view of minimizing the total loss due to multi-term loss functional by a series of densely connected neural networks which are also referred to as feed-forward deep neural networks. To illustrate the effectiveness and usages of our proposed approach, four computational examples are given. We compared the proposed scheme with the analytical or true solution and other methods existing in the literature. The findings indicate that the suggested approach provides more accurate results and is an efficient deep learning method, offering a reliable and accurate estimation for both forward and inverse problems of multi-dimensional non-linear Klein-Gordon equations. Therefore, the proposed computational approach is efficient in solving challenging physical forward and inverse problems with different boundary and initial conditions.