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<p>This work presents a modified physics informed neural network (PINN) framework that integrates gradient augmentation and an adaptive weighted loss to solve heat conduction problems with time-dependent thermal conductivity and source term. The method uses automatic differentiation to compute solution gradients, which are incorporated as auxiliary outputs to improve training smoothness and accuracy. An adaptive weighted loss mechanism dynamically balances the contributions of the governing equation residual, initial conditions, and boundary conditions during training, thereby enhancing convergence, stability, and robustness. The proposed adaptive loss weighted gradient-enhanced PINN framework is validated on a benchmark problem, both with and without a known analytical solution and its performance is compared against a standard PINN using fixed loss weights. Results demonstrate that the proposed approach achieve superior accuracy, even when trained with fewer collocation points. To further assess robustness and stability, a statistical error analysis based on variance and standard deviation was conducted. The findings indicate that the proposed method provides a reliable and scalable computational framework for variable-coefficient heat conduction problems in nonlinear field theories.</p>

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Gradient enhanced physics informed neural networks with adaptive loss for 2D heat equations with variable and non smooth conductivity

  • 期刊论文

  • 出版模式：

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  • 发表日期：

    2026年5月26日

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  • Meseret Cherkos Tessema,
  • Alemayehu Tamirie Deresse,
  • Mitiku Daba Firdi,
  • Tamirat Temesgen Dufera

摘要

This work presents a modified physics informed neural network (PINN) framework that integrates gradient augmentation and an adaptive weighted loss to solve heat conduction problems with time-dependent thermal conductivity and source term. The method uses automatic differentiation to compute solution gradients, which are incorporated as auxiliary outputs to improve training smoothness and accuracy. An adaptive weighted loss mechanism dynamically balances the contributions of the governing equation residual, initial conditions, and boundary conditions during training, thereby enhancing convergence, stability, and robustness. The proposed adaptive loss weighted gradient-enhanced PINN framework is validated on a benchmark problem, both with and without a known analytical solution and its performance is compared against a standard PINN using fixed loss weights. Results demonstrate that the proposed approach achieve superior accuracy, even when trained with fewer collocation points. To further assess robustness and stability, a statistical error analysis based on variance and standard deviation was conducted. The findings indicate that the proposed method provides a reliable and scalable computational framework for variable-coefficient heat conduction problems in nonlinear field theories.