Introductory Notes on Learning$^2$

תקציר מקורי באנגליתarXiv:2609.06546v1 Announce Type: cross Abstract: Although machine learning can be used to predict the evolution of physical systems from data, a formulation that learns only the system state at each time leaves the temporal and dynamical structure of the solution to be resolved within a broad hypothesis space. We introduce Learning$^2$, a representation-level framework that structures this space by coupling a primary representation to a second representation through a known physical transformation. The resulting cross-representation constraint restricts the effective hypothesis space and provides an ante-hoc, physically interpretable criterion for excluding solutions that satisfy the primary representation alone. We instantiate Learning$^2$ through EuLaNet, an Eulerian--Lagrangian represe