Abstract
Extracting governing equations and dynamics from data is crucial for prediction, sensing and control of resource-constrained engineering systems. Identifying the fewest possible model terms or sensor measurements often translates into sparsity penalties on optimization design variables. Current approaches rely on relaxation, heuristics, and trial-and-error selection of hyperparameters, which challenge the interpretability and verification of resulting models. In this talk, we propose a statistical mechanics approach for robust sparse equation discovery, using free energies to analyze the optimization landscape, optimize hyperparameters and quantify uncertainty with respect to noise and data volume. We illustrate how this perspective adapts to optimal sensor placement, providing optimization landscapes and critical noise regimes for reconstruction of high-dimensional fields from sparse sensors and data-driven priors. We showcase how these tools can be used for constrained optimization of sensor placement in nuclear digital twins.