We introduce the neural empirical interpolation method (NEIM), a neural network-based alternative to the discrete empirical interpolation method (DEIM) for reducing the time complexity of computing nonlinearities in reduced order models (ROMs) for parameterized nonlinear partial differential equations. NEIM accomplishes this by approximating an affine decomposition of the nonlinear term of the ROM, where the vector terms of the expansion are given by neural networks depending on the ROM solution and the coefficients are given by an interpolation of some “optimal” coefficients. The approximate affine decomposition is obtained via a greedy selection of parameters for which current approximations have large error. Thus, NEIM can be viewed as a greedy method of training a DeepONet for nonlinear terms in ROMs. Unlike DEIM, the NEIM algorithm is a data-driven method. It is also distinguished by its being greedy in parameter space and being efficient for both local and nonlocal nonlinearities, unlike DEIM which works well for component-wise nonlinearities. We compare the two algorithms in numerical experiments with finite element methods and physics-informed neural networks and show that by using NEIM, we can achieve comparable error to DEIM despite differences in the offline phase of the two methods.