In our prior work [10], neural networks with local converging inputs (NNLCI) were introduced for solving one-dimensional conservation equations. Two solutions of a conservation law in a converging sequence, computed from low-cost numerical schemes, and in a local domain of dependence of the space-time location, were used as the input to a neural network in order to predict a high-fidelity solution at a given space-time location. In the present work, we extend the method to two-dimensional conservation systems and introduce different solution techniques. Numerical results demonstrate the validity and effectiveness of the NNLCI method for application to multi-dimensional problems. In spite of low-cost smeared input data, the NNLCI method is capable of accurately predicting shocks, contact discontinuities, and the smooth region of the entire field. The NNLCI method is relatively easy to train because of the use of local solvers. The computing time saving is between one and two orders of magnitude compared with the corresponding high-fidelity schemes for two-dimensional Riemann problems. The relative efficiency of the NNLCI method is expected to be substantially greater for problems with higher spatial dimensions or smooth solutions.