High-fidelity simulations of engineering systems can yield accurate results but are often prohibitively expensive for optimization processes and real-time applications. Deep learning approaches offer superior inference speeds but lack theoretical guarantees when operating on unseen data due to their black box nature. Recent advances in scientific machine learning present hybrid approaches that embed governing physical equations into statistical methods. SINDy (sparse identification of nonlinear dynamics) [1] has demonstrated remarkable capabilities in identifying governing equations from data in a range of applications through sparse regression, whilst retaining interpretability and generalisability. However, SINDy scales combinatorially with the number of states in the measurement data, and its successful application relies on some prior knowledge of the system’s underlying equation structure. Recent developments of hybrid methods that combine SINDy with deep learning [2, 3] present novel approaches in learning large systems on a lower-dimensional manifold. This formulation enables stable forward solving of high-dimensional spatio-temporal systems entirely within the latent space. We explore and compare these approaches with other leading scientific machine learning methods, such as PDE solvers and operator learning type methods [4], evaluating their ability to predict temporal evolution on the PDEBench suite [5]. Finally, we explore this approach in aeroelastic applications, learning parameterised equations to quantify the safe operational envelope limited by limit cycle oscillation and flutter boundaries.