Physics-informed neural networks with parameter asymptotic strategy for learning singularly perturbed convection-dominated problem

Physics-informed neural networks (PINN) have proven their effectiveness in solving partial differential equations (PDEs). Nevertheless, existing networks cannot model finely detailed signals and therefore fail to represent a signal's spatial and temporal derivatives. Traditional PINN are unable to approximate solutions of singularly perturbed boundary value problems that have solutions exhibiting sharp boundary layers and steep gradients with sufficient accuracy. A new asymptotic parameter PINN (PAPINN) is proposed to solve singular perturbation-dominated problems. This approach approximates the smooth solution by optimizing the neural network with large perturbation parameters, which are then used as initial values of the neural network with small perturbation parameters to approximate the singular solution. The method avoids the disadvantages of uncertainty of random parameters and manual setting of initial weights, and gives the network better initial weights. It offers a feasible deep learning approach for solving the singular perturbation problem without requiring a priori boundary layer information. By solving numerical examples of convection-diffusion equations originating from magnetic fluids, the accuracy and convergence efficiency of this method are compared with those of PINN and gPINN. The results show that the method can effectively approximate the large gradient solution of the convection-dominated diffusion equation with an accuracy of order 10-3 and has better convergence speed and stability than PINN and gPINN.