Many engineering applications involve oscillatory systems that are both difficult to model and to observe. This is the case of micro electro-mechanical systems (MEMS), which are typically observed through indirect measurements such as capacitive readings. Developing accurate predictive models of their dynamics can unlock new possibilities for the monitoring and control of engineering systems [1]. However, constructing reliable dynamical models remains challenging, as purely data-driven methods often struggle to generalise beyond the training data, while physics-based model may be biased by missing physics or excessive complexity. Physics-enhanced, data-driven model discovery methods, like the Sparse Identification of Nonlinear Dynamics (SINDy) method [2], aim to mitigate these limitations by combining the strengths of both approaches. Models identified via SINDy are expressed as sparse combination of simple functional terms, enabling clear physical interpretability. In practice, however, available measurements are frequently partial or indirect. To lift the dimensionality of these measurements suitably for dynamic modelling, Time-Delay Embedding (TDE) can be employed [3]. Within the resulting time-delay coordinate system, SINDy can then be applied. Nevertheless, the time-delay coordinate system does not generally yield sparse, and thus physically interpretable, models [2]. To address this issue, a Normal Form (NF) transformation is applied to simplify the dynamics as much as possible [3,4]. In this work, applications to MEMS are presented to validate the combined use of SINDy, TDE, and NF transformation for modelling indirectly observed dynamics. [1] L. Rosafalco et al. EKF-SINDy: Empowering the extended Kalman filter with sparse identification of nonlinear dynamics. Computer Methods in Applied Mechanics and Engineering, Vol. 431, 117264, 2024. [2] S.L. Brunton et al. Discovering governing equations from data by sparse identification of nonlinear dynamical systems. Proceedings of the National Academy of Sciences, Vol. 113 (15), 3932–3937, 2016. [3] M. Cenedese et al. Data-driven modeling and prediction of non-linearizable dynamics via spectral submanifolds. Nature Communications, Vol. 13, 872, 2022. [4] A. Vizzaccaro et al. Direct parametrisation of invariant manifolds for non-autonomous forced systems including superharmonic resonances. Nonlinear Dynamics, Vol. 112, 6255–6290, 2024.