A core challenge in scientific machine learning is how to learn from limited, noisy, or strategically chosen data. This talk presents the mathematics of sparse sensing—the problem of choosing maximally informative measurements for high-dimensional estimation and control. Sensor placement defines a highly nonconvex optimization landscape and, in general, is NP-hard. Existing methods provide limited insight into the underlying landscape of sensing objectives.

We address this by characterizing the sensing landscape explicitly through a statistical mechanics formulation, deriving a Hamiltonian whose 1-body and 2-body terms are computed directly from training data. At each greedy iteration, this yields a spatial energy landscape revealing how information gain varies across candidate sensor locations, directly inspiring scalable placement algorithms. The framework unifies ideas from Bayesian inference, D-optimal design, and statistical physics, and yields quantitative reconstruction guarantees: explicit bounds connecting sensor budget, noise level, and basis rank to estimation error. The resulting methods enable robust field reconstruction, a priori uncertainty quantification, and constrained sensor placement in safety-critical environments. These ideas further extend to nonlinear system identification, where sparse discovery of governing equations (SINDy) can be interpreted as a learning problem with an underlying energy landscape.