Scientific machine learning for large-scale engineering systems requires models that are sample-efficient, interpretable, and explicitly constrained by physics and uncertainty. This talk presents new mathematical and algorithmic foundations for sparse sensing and nonlinear system identification in high-dimensional dynamical systems, with an emphasis on scalability and performance guarantees. We develop a statistical-mechanical formulation of the sensor placement problem, deriving an energy landscape (Hamiltonian) whose minima correspond to optimally informative sensor configurations under uncertainty and physical constraints. This perspective combines Bayesian inference with a data-driven prior and a greedy energy minimizing placement algorithm, yielding per-iteration spatial optimization landscapes, enabling real-time adaptive sensing and online reconfiguration. Building on these ideas, we show how physics-informed embeddings enable reliable state estimation, uncertainty quantification, and closed-loop prediction in safety-critical aerospace and nuclear systems where black-box ML fails. Finally, we highlight connections to nonlinear system identification—where sparse discovery of governing equations (SINDy) is recast as a statistical-mechanical inference problem—providing insights into identifiability, stability, and long-horizon generalization. Applications include large-scale fluid flows, flexible structures, and nuclear energy subsystems, demonstrating improved reconstruction accuracy, robust uncertainty quantification, and substantial reductions in sensing and computational cost.