Abstract: The e^N method based on linear stability theory (LST) is one of the more reliable methods in the prediction of boundary layer transition. In order to greatly simplify and automate the solution process of the traditional LST eigenvalue problem, the convolutional neural network (CNN) is trained on the LST analysis sample set of the boundary layer similarity solution. For the streamwise and crossflow instabilities, the local growth rate, N factor and transition location are predicted by CNN on a naturally laminar airfoil and an infinite swept-back wing respectively, which are in good agreement with the results of standard LST. It is verified that CNN can encode the velocity derivative information of the boundary layer profile into a scalar feature that satisfies the Galilean invariance, and plays a role in characterizing the pressure gradient in the boundary layer of an airfoil or the crossflow intensity in the boundary layer of a swept-back wing. Based on the prediction of LST eigenvalues by CNN, the total loss function is constructed by the governing equations of LST, the boundary conditions and the trivial solution penalty term to train the physics-informed neural network (PINN), which realizes an accurate prediction of LST eigenfunctions without relying on samples. The results show that the PINN model can provide an effective modeling method for the eigenfunction problem of LST.