The solution to partial differential equations based on scientific machine learning approaches, such as the physics-informed neural networks (PINNs) method, the deep Ritz method, or the weak adversarial networks method, has shown in recent years encouraging results for several classes of initial and boundary-value problems. However, their ability to surpass, particularly in terms of accuracy, classical discretization methods remains a significant challenge. In this talk, we will describe how a multilevel neural networks approach can help reduce the numerical error for deep learning methods. The main idea consists in computing an initial approximation to the problem using a simple neural network and in estimating, in an iterative manner, a correction by solving the problems for the residual errors with new networks of increasing complexity. This sequential process allows one to significantly decrease the error in approximations of linear and nonlinear problems and achieve, in some cases, machine precision. We will also show how the multilevel method can be used to estimate errors in quantities of interest. Numerical examples in 1D and 2D are presented to demonstrate the effectiveness of the proposed approach using PINNs.