Matched Asymptotic Expansions‐Based Transferable Neural Networks for Singular Perturbation Problems

ABSTRACT In this paper, by utilizing the theory of matched asymptotic expansions, an efficient and accurate neural network method, named as “MAE‐TransNet,” is developed for solving singular perturbation problems in general dimensions, whose solutions usually change drastically in some narrow boundary layers. The TransNet is a two‐layer neural network with specially pretrained hidden‐layer neurons. In the proposed MAE‐TransNet, the inner and outer solutions produced from the matched asymptotic expansions are first approximated by a TransNet with nonuniform hidden‐layer neurons and a TransNet with uniform hidden‐layer neurons, respectively. Then, these two solutions are combined with a matching term to obtain the composite solution, which approximates the asymptotic expansion solution of the singular perturbation problem. This process enables the MAE‐TransNet method to retain the precision of the matched asymptotic expansions while maintaining the efficiency and accuracy of TransNet. Meanwhile, the rescaling of the sharp region allows the same pretrained network parameters to be applied to boundary layers with various thicknesses, thereby improving the transferability of the method. Notably, for coupled boundary layer problems, a computational framework based on MAE‐TransNet is also constructed to effectively address issues resulting from the lack of relevant matched asymptotic expansion theory in such problems. Our MAE‐TransNet is thoroughly compared with TransNet, PINN, and Boundary‐Layer PINN (BL‐PINN) on various benchmark problems, including 1D linear and nonlinear problems with boundary layers, the 2D Couette flow problem, a 2D coupled boundary layer problem, and the 3D Burgers vortex problem. Numerical results demonstrate that MAE‐TransNet significantly outperforms other neural network methods in capturing the characteristics of boundary layers, improving the accuracy, and reducing the computational cost.