  Given observations of a physical system, identifying the underlying
non-linear governing equation is a fundamental task, necessary both for gaining
understanding and generating deterministic future predictions. Of most
practical relevance are automated approaches to theory building that scale
efficiently for complex systems with many degrees of freedom. To date,
available scalable methods aim at a data-driven interpolation, without
exploiting or offering insight into fundamental underlying physical principles,
such as locality of interactions. In this work, we show that various physical
constraints can be captured via tensor network based parameterizations for the
governing equation, which naturally ensures scalability. In addition to
providing analytic results motivating the use of such models for realistic
physical systems, we demonstrate that efficient rank-adaptive optimization
algorithms can be used to learn optimal tensor network models without requiring
a~priori knowledge of the exact tensor ranks. As such, we provide a
physics-informed approach to recovering structured dynamical laws from data,
which adaptively balances the need for expressivity and scalability.
