Geometry-informed neural architectures for spatiotemporal sensing and spaceborne autonomy: operator-embedded and manifold-aware spectral learning frameworks

Modern spaceborne sensing and autonomous systems increasingly rely on deep learning methods for tasks, such as: reconstruction, classification, object extraction, tracking, and dynamical modeling. However, conventional, state-of-the-art neural architectures generally treat geometry as an emergent property, inferred from data rather than as a structural prior. This often leads to models that are physically inconsistent, too reliant on data volume, and difficult to interpret and validate in safety-critical aerospace applications. This thesis develops a unified framework for \textbf{geometry-informed neural architectures}, where spatial, spectral, functional, and temporal geometrical priors are directly embedded into learning systems. The central hypothesis is that incorporating geometry, such as: differential operators, spectral representations, and manifold-aware constraints within neural architectures improves robustness, physical consistency, and physics-consistent generalization in spaceborne sensing and autonomy tasks. The thesis presents four principal contributions. First, a \textit{spatial geometry-informed reconstruction framework} is introduced for soil moisture field estimation from sparse satellite observations. Fixed, discrete differential operators and geometrical structures are embedded into a neural model, enabling edge-aware interpolation and improved, high-definition reconstruction fidelity without excessive parameterization. Second, geometry is extended from reconstruction to \textit{object extraction, i.e. objectization, and temporal reasoning}. A spectral pipeline based on Laplace-Beltrami embeddings is developed to represent reconstructed geophysical fields as coherent-across-temporal-domain geometric objects. This formulation enables geometry-consistent feature tracking across time and preserves intrinsic shape structure. Third, a novel framework for \textit{geometry-informed autonomous dynamics modeling} is proposed through Laplacian-Spectral Dynamic Movement Primitives - LSDMPs. By expanding trajectory forcing terms in the eigenbasis of a temporal graph Laplacian, orbital dynamics modeling is cast as a geometry-aware spectral approximation problem. This establishes a unified operator-theoretic perspective connecting spatial Laplace-Beltrami embeddings and temporal Laplacian representations. Finally, the work develops geometry embedding to the \textit{functional and Hilbert-space domain} for remote sensing image classification. Instead of relying only on learned convolutional filters, operator-defined mappings project image patches into high-dimensional functional spaces, enforcing inner-product structure and promoting structured feature representations. The dissertation demonstrates that embedding geometry explicitly - rather than learning it implicitly - provides a reliable method towards more physics-consistent and stable neural systems for spaceborne applications - with a particular attention to remote sensing. The resulting framework bridges differential geometry, spectral graph theory, and deep learning, and defines geometry-informed neural designs as a viable paradigm for next-generation remote sensing and autonomous aerospace systems.