In this work, we explore recent advances in scientific machine learning for the simulation and analysis of complex fluid flows. Our main motivation is to enhance well-established numerical solvers by integrating data-driven, physics-informed and deep learning methodologies for unsteady viscoelastic flows. Two representative problems are investigated: confined benchmarks and free surface flows. For confined benchmarks, we focus on learning reduced latent representations based on energy-driven dimensionality reduction incorporating non-Newtonian physics—such as elastic stresses and constitutive behavior [1]. Adopting different viscoelastic fluids, we show that we can model the dynamics of the reduced order model using SINDy with an appropriate choice of kernel and provide better models than the ones with linear PCA. For free surface flows, we investigate deep learning approaches to predict droplet shape evolution and morphological changes under complex flow conditions \cite{Paper2}. Preliminary results show that video-based models incorporating attention mechanisms provide more consistent predictions of droplet evolution and improved accuracy in long-horizon forecasting compared to simpler baseline models. These results highlight the importance of spatial-temporal modeling for droplet dynamics prediction. This work provides insight into how machine learning techniques originally developed for Newtonian fluid mechanics can be systematically extended to non-Newtonian flows, where the interplay between viscous and elastic effects introduces additional levels of complexity.