This talk presents a family of physics-informed statistical learning methods for spatial and functional data, aimed at integrating data-driven modeling with scientific knowledge of the phenomena under study. The proposed approaches extend classical nonparametric and semiparametric methods by incorporating regularization terms based on differential operators, ranging from simple Laplacians to problem-specific, possibly nonlinear, partial differential equations. These penalties encode information about the underlying physical processes while accounting for the geometry of the observation domain, enabling the analysis of data defined over non-convex multidimensional regions, curved surfaces, and networks. The resulting framework combines the flexibility of statistical modeling with the expressive power of differential equations, giving rise to new theoretical, inferential, and computational challenges. I will also illustrate how differential regularization can be incorporated into functional dimension reduction methods, such as functional principal component analysis and functional partial least squares regression. Applications from the life and environmental sciences will demonstrate the potential of these methods for analyzing complex, high-dimensional data.