We propose a Neural Evolutionary Kernel Method (NEKM), for solving a class of time-dependent partial differential equations (PDEs) via deep neural network (DNN)-based kernel representations. By integrating boundary integral techniques with operator learning, neural network architectures are designed to predict solutions of time-dependent partial differential equations (PDEs) at each time step, while embedding prior mathematical structure to enhance both computational efficiency and solution accuracy. Numerical experiments on the heat, wave, and Schr{o}dinger equations demonstrate that the Neural Evolutionary Kernel Method (NEKM) achieves high accuracy and favorable computational efficiency.
