Vector field design plays a fundamental role in a wide range of robot motion control and navigation strategies. More recently, effort has been made to learn these vector fields from data rather than rely on first-principles formulation. Acquiring data necessary to construct these vector fields, whether from robot exploration or expert demonstration, is expensive. This means we must rely on sparse data to build them. However, generating reliable, dense representations from this limited information remains a fundamental challenge. Current data-driven methods treat vector field reconstruction as a localized regression problem. By optimizing primarily for local kinematics, they lack the structural inductive bias needed to preserve the topological structure, frequently generating spurious attractors that fundamentally compromise reliability in unobserved regions. To close this gap, this dissertation introduces a structured operator-learning framework that lifts the state space via learnable basis functions to identify a linear Koopman operator. Because this operator governs the global dynamics, it ensures the vector field remains structurally consistent and well-behaved across unobserved regions. The framework is designed for modularity: it can incorporate physics-informed losses, task-specific lifting functions, and ensemble modeling for active flow discovery. The framework’s efficacy is demonstrated across a spectrum of robot motion planning tasks. For informative path planning, it enables active sensing to autonomously reconstruct unknown flows. For reactive navigation, it generates stable vector fields from demonstrations and obstacle-avoiding flows in complex geometries. Overall, this work establishes that operator-theoretic priors provide the global structure necessary for reliable robot navigation.