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<p>This paper proposes the Theoretically Informed Periodic PINN (TI-PerPINN) to resolve the boundary layer issues on circular domains while solving convection diffusion problems in two dimensions. In contrast to other research, this article aims to predict the solution without a prior knowledge of the layer using a neural network architecture. TI-PerPINN leverages a boundary-fitted polar-coordinate transformation to naturally align the computational domain with the transform domain. We proposed two composite loss components that integrate periodic boundary conditions in the novel framework. Furthermore, the framework employs an adaptive dual-tanh activation function and a robust Huber loss to mitigate vanishing gradients and the influence of outliers during training. The theoretical motivation for constructing loss functions is presented with error analysis. Numerical experiments further demonstrate the efficiency of the theoretical aspect of TI-PerPINN. The model investigated here is of significant importance in the fields of thermo-hydraulics and fluid mechanics.</p>

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TI-PerPINN: a theoretically guided neural network driven approaches for singularly perturbed convection dominated differential equations on circular domains

  • 期刊论文

  • 发表日期：

    2026年6月1日

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  • Subhendu Maity,
  • Krishna Rathore

摘要

This paper proposes the Theoretically Informed Periodic PINN (TI-PerPINN) to resolve the boundary layer issues on circular domains while solving convection diffusion problems in two dimensions. In contrast to other research, this article aims to predict the solution without a prior knowledge of the layer using a neural network architecture. TI-PerPINN leverages a boundary-fitted polar-coordinate transformation to naturally align the computational domain with the transform domain. We proposed two composite loss components that integrate periodic boundary conditions in the novel framework. Furthermore, the framework employs an adaptive dual-tanh activation function and a robust Huber loss to mitigate vanishing gradients and the influence of outliers during training. The theoretical motivation for constructing loss functions is presented with error analysis. Numerical experiments further demonstrate the efficiency of the theoretical aspect of TI-PerPINN. The model investigated here is of significant importance in the fields of thermo-hydraulics and fluid mechanics.