  We propose a statistical learning framework based on group-sparse regression
that can be used to 1) enforce conservation laws, 2) ensure model equivalence,
and 3) guarantee symmetries when learning or inferring differential-equation
models from measurement data. Directly learning $\textit{interpretable}$
mathematical models from data has emerged as a valuable modeling approach.
However, in areas like biology, high noise levels, sensor-induced correlations,
and strong inter-system variability can render data-driven models nonsensical
or physically inconsistent without additional constraints on the model
structure. Hence, it is important to leverage $\textit{prior}$ knowledge from
physical principles to learn "biologically plausible and physically consistent"
models rather than models that simply fit the data best. We present a novel
group Iterative Hard Thresholding (gIHT) algorithm and use stability selection
to infer physically consistent models with minimal parameter tuning. We show
several applications from systems biology that demonstrate the benefits of
enforcing $\textit{priors}$ in data-driven modeling.
