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This paper discusses a physics-informed methodology aimed at reconstructing efficiently the fluid state of a system. Herein, the generation of an accurate reduced order model of two-dimensional unsteady flows from data leverages on sparsity-promoting statistical learning techniques. The cornerstone of the approach is l1regularised regression, resulting in sparsely-connected models where only the important quadratic interactions between modes are retained. The original dynamical behaviour is reproduced at low computational costs, as few quadratic interactions need to be evaluated. The approach has two key features. First, interactions are selected systematically as a solution of a convex optimisation problem and no a priori assumptions on the physics of the flow are required. Second, the presence of a regularisation term improves the predictive performance of the original model, generally affected by noise and poor data quality. Test cases are for two-dimensional lid-driven cavity flows, at three values of the Reynolds number for which the motion is chaotic and energy interactions are scattered across the spectrum. It is found that: (A) the sparsification generates models maintaining the original accuracy level but with a lower number of active coefficients; this becomes more pronounced for increasing Reynolds numbers suggesting that extension of these techniques to real-life flow configurations is possible; (B) sparse models maintain a good temporal stability for predictions. The methodology is ready for more complex applications without modifications of the underlying theory, and the integration into a cyber-physical model is feasible.