Abstract: Deep learning based methods are being employed to resolve the computational challenges of high-dimensional partial differential equations(PDEs). But the computation of the high order derivatives of neural networks is costly, and lack robustness. We propose the local deep learning method by simultaneously approximating the function value and derivatives. Moreover, a priori error estimate of the method is derived by analyzing the approximation error and the generalization error. Numerical examples are carried out to demonstrate that local deep learning method is efficient, robust, flexible, and is particularly well-suited for high-dimensional PDEs with high order derivatives.