In this work, we design an iteration-based unsupervised neural network algorithm and the theoretical bound of the loss function through Barron space to solve reaction–convection–diffusion-based magnetohydrodynamic (MHD) coupled systems for 1D and 2D problems. Our algorithm is also be able to capture the presence of boundary layers. In general, these systems are characterized by strong coupling in the reaction and convection processes and involve non-diagonally dominant matrices in the context of convection coupling. Traditional numerical techniques face challenges in approximating these problems due to the failure of the required maximum principle, which is used for the well-posedness and further convergence analysis of numerical solutions. We specifically provide a new modified iterative physics-informed neural network (MI-PINN)-based unsupervised deep learning algorithm to capture the layer behavior of singularly perturbed strongly coupled steady-state problems, appearing in MHD flows where theoretical analysis and numerical methods are limited. A different analysis based on the sigmoid activation function is provided for the steady-state case, which shows that the empirical loss under the $ L^2 $ norm is bounded and converges for the two layer-based networks- whenever the solution lies in the Barron space. Additionally, the proposed algorithm improves the neural network's output without using the boundary layer functions a priori and does not use hard constraints or interpolation with the neural network's solution. The experimental results show that the proposed algorithm performs very well for MHD flows appearing in the form of strongly coupled systems.
Citation: Arihant Patawari, Shridhar Kumar, Pratibhamoy Das. A modified iterative PINN algorithm for strongly coupled system of boundary layer originated convection diffusion reaction problems in MHD flows with analysis[J]. Networks and Heterogeneous Media, 2025, 20(3): 1026-1060. doi: 10.3934/nhm.2025045
In this work, we design an iteration-based unsupervised neural network algorithm and the theoretical bound of the loss function through Barron space to solve reaction–convection–diffusion-based magnetohydrodynamic (MHD) coupled systems for 1D and 2D problems. Our algorithm is also be able to capture the presence of boundary layers. In general, these systems are characterized by strong coupling in the reaction and convection processes and involve non-diagonally dominant matrices in the context of convection coupling. Traditional numerical techniques face challenges in approximating these problems due to the failure of the required maximum principle, which is used for the well-posedness and further convergence analysis of numerical solutions. We specifically provide a new modified iterative physics-informed neural network (MI-PINN)-based unsupervised deep learning algorithm to capture the layer behavior of singularly perturbed strongly coupled steady-state problems, appearing in MHD flows where theoretical analysis and numerical methods are limited. A different analysis based on the sigmoid activation function is provided for the steady-state case, which shows that the empirical loss under the $ L^2 $ norm is bounded and converges for the two layer-based networks- whenever the solution lies in the Barron space. Additionally, the proposed algorithm improves the neural network's output without using the boundary layer functions a priori and does not use hard constraints or interpolation with the neural network's solution. The experimental results show that the proposed algorithm performs very well for MHD flows appearing in the form of strongly coupled systems.
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Arihant Patawari, Shridhar Kumar, Pratibhamoy Das. A modified iterative PINN algorithm for strongly coupled system of boundary layer originated convection diffusion reaction problems in MHD flows with analysis[J]. Networks and Heterogeneous Media, 2025, 20(3): 1026-1060. doi: 10.3934/nhm.2025045
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