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There has been a lot of interest recently in leveraging machine learning approaches for modeling and analysis in the sciences and engineering. This poses significant challenges and requirements related to data efficiency, interpretability, and robustness. For scientific problems there is often a lot of prior knowledge about general underlying physical principles, existence of low dimensional latent structures, or groups of invariances or equivariances. We discuss approaches for representing some of this knowledge to enhance learning methods by using results on manifold embeddings, stochastic processes within manifolds, and harmonic analysis. We show how the approaches can be used for high-dimensional stochastic dynamical systems with slow-fast time-scale separations to learn from observations, slow variable representations and reduced models for the dynamics. We also discuss a few other examples where utilizing geometric structure has the potential to improve outcomes in scientific machine learning.

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