We present the localized discrete empirical interpolation method (LDEIM) for constructing reduced-order models based on proper orthogonal decomposition for PDEs with nonlinear terms. As opposed to standard DEIM, where the nonlinear term is projected onto one global subspace, our localized approach computes and projects onto several local spaces, each tailored to a particular region of characteristic system behavior. In the offline computational phase, LDEIM uses machine learning tools such as feature extraction and clustering to learn these regions directly from system states and then constructs the corresponding subspaces. In the online computational phase, machine-learning-based procedures classify the current system state into one of the learned regimes as the computation proceeds. The dimensions of the local DEIM subspaces, and thus the computational costs, remain low even though the system might exhibit a wide range of different behaviors as it passes through the state space. We demonstrate our approach for a reacting flow example of an H2-Air flame.