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A Second-Order Network Structure Based on Gradient-Enhanced Physics-Informed Neural Networks for Solving Parabolic

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Summary

Physics-informed neural networks (PINNs) solve partial differential equations (PDEs) by integrating them into neural network loss functions. This study introduces a novel second-order neural network structure and advanced techniques to enhance PINN performance for complex wave equations.

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deep mixed residual methodparabolic partial differential equationsphysics-informed neural networkssecond-order neural network

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Area of Science:

  • Computational physics
  • Applied mathematics
  • Machine learning for scientific computing

Background:

  • Physics-informed neural networks (PINNs) effectively solve forward and inverse partial differential equation (PDE) problems by embedding physical laws into neural network loss functions.
  • PINNs offer a powerful framework for scientific discovery but can face challenges with complex equations and computational cost.

Purpose of the Study:

  • To introduce a novel neural network structure for solving parametric wave equations.
  • To enhance the performance and efficiency of PINNs using adaptive strategies and advanced computational methods.
  • To validate the proposed method on nonlinear parabolic partial differential equations.

Main Methods:

  • A parametric light wave equation was discretized using the central difference method.
  • A novel second-order neural network structure was developed based on the discretization scheme.
  • Adaptive activation function and gradient-enhanced strategies were employed to boost neural network performance.
  • The deep mixed residual method (MIM) was utilized to mitigate high computational costs associated with gradient enhancement.

Main Results:

  • The developed second-order neural network structure, combined with adaptive and gradient-enhanced strategies, demonstrated improved performance in solving PDEs.
  • The deep mixed residual method effectively reduced computational overhead while maintaining accuracy.
  • Numerical examples involving nonlinear parabolic PDEs confirmed the method's effectiveness and robustness.

Conclusions:

  • The proposed second-order neural network structure offers a promising advancement for PINNs.
  • The integration of adaptive strategies and the deep mixed residual method enhances the efficiency and applicability of PINNs for complex scientific problems.
  • This approach provides a viable computational tool for solving challenging partial differential equations in various scientific domains.