A key enabling aspect for digital twins is the ability to accurately model physical processes in real-time. When high-fidelity models are created using, e.g., finite element analysis (FEA) software, model reduction is required to achieve accurate and compact models capable of real-time execution. Although many model reduction techniques rely on precise knowledge of the mathematical model, the semi-discretized operators are typically not readily available when working with (commercial) FEA software. Motivated by this challenge, operator inference was introduced in [2]. Operator inference non-intrusively constructs a dynamic model using only simulation snapshots. However, guaranteeing stability of the nonlinear model is a challenge. Stability guarantees based on Lyapunov analysis are given in [3]. However, the stability results are only valid for a bounded subset of state space and for cubic nonlinearities. Inspired by the sparsity-promoting hyperreduction technique of [1], we propose an extension of operator inference. In our extension, we can simultaneously guarantee global asymptotic stability and efficiently treat systems with polynomial nonlinearities of arbitrary degree. The method paves the way for flexible and reliable digital twins of dynamic systems involving nonlinear structural mechanics. We demonstrate the proposed method with numerical examples of systems involving large displacement and nonlinear material laws.