  We present a learning algorithm for discovering conservation laws given as
sums of geometrically local observables in quantum dynamics. This includes
conserved quantities that arise from local and global symmetries in closed and
open quantum many-body systems. The algorithm combines the classical shadow
formalism for estimating expectation values of observable and data analysis
techniques based on singular value decompositions and robust polynomial
interpolation to discover all such conservation laws in unknown quantum
dynamics with rigorous performance guarantees. Our method can be directly
realized in quantum experiments, which we illustrate with numerical
simulations, using closed and open quantum system dynamics in a
$\mathbb{Z}_2$-gauge theory and in many-body localized spin-chains.
