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Efficiently reducing models of chemically reacting flows is often challenging because their characteristic features such as sharp gradients in the flow fields and couplings over various time and length scales lead to dynamics that evolve in high-dimensional spaces. In this work, we show that online adaptive reduced models that construct nonlinear approximations by adapting low-dimensional subspaces over time can predict well latent dynamics with properties similar to those found in chemically reacting flows. The adaptation of the subspaces is driven by the online adaptive empirical interpolation method, which takes sparse residual evaluations of the full model to compute low-rank basis updates of the subspaces. Numerical experiments with a premixed flame model problem show that reduced models based on online adaptive empirical interpolation accurately predict flame dynamics far outside of the training regime and in regimes where traditional static reduced models, which keep reduced spaces fixed over time and so provide only linear approximations of latent dynamics, fail to make meaningful predictions.