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We develop a data-driven method, based on semi-supervised classification, to predict the asymptotic state of multistable systems when only sparse spatial measurements of the system are feasible. Our method predicts the asymptotic behavior of an observed state by quantifying its proximity to the states in a precomputed library of data. To quantify this proximity, we introduce a sparsity-promoting metric-learning (SPML) optimization, which learns a metric directly from the precomputed data. The optimization problem is designed so that the resulting optimal metric satisfies two important properties: (i) It is compatible with the precomputed library, and (ii) It is computable from sparse measurements. We prove that the proposed SPML optimization is convex, its minimizer is non-degenerate, and it is equivariant with respect to scaling of the constraints. We demonstrate the application of this method on two multistable systems: a reaction-diffusion equation, arising in pattern formation, which has four asymptotically stable steady states and a FitzHugh-Nagumo model with two asymptotically stable steady states. Classifications of the multistable reaction-diffusion equation based on SPML predict the asymptotic behavior of initial conditions based on two-point measurements with 95% accuracy when moderate number of labeled data are used. For the FitzHugh-Nagumo, SPML predicts the asymptotic behavior of initial conditions from one-point measurements with 90% accuracy. The learned optimal metric also determines where the measurements need to be made to ensure accurate predictions.