Abstract: To address the issue of insufficient accuracy when solving convection-dominated diffusion equations with drastic gradient changes near the boundary layer using Physics-Informed Neural Networks (PINNs), a neural network solution method incorporating a parameter progressive strategy is proposed. This method initially approximates the smooth solution of the large diffusion parameter equation, then progressively reduces the diffusion parameter while using the optimal network parameters from the large diffusion parameter as the initial values for training the small diffusion parameter neural network. By iteratively optimizing the Physics-Informed Neural Networks through parameter cycling, the neural network’s representational capacity is enhanced, thereby improving the approximation accuracy of the convection-dominated diffusion problem. And finally a high-precision singular solution for the small diffusion parameter is obtained. A comparison of the accuracy and convergence efficiency of present method with those of PINNs and gPINNs shows that this method can efficiently approximate the large gradient solution of the convective dominant diffusion equation with an accuracy of the order of 10^-3 under the condition of unknown boundary layer position. Meanwhile, the present method has more advantages and better performance than PINNs and gPINNs in terms of convergence speed and stability.

Key words: diffusion equation, boundary layer, physical information neural networks, deep learning, convective diffusion