Introductory Notes on Learning$^2$

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 representation for fluid dynamics. Given the velocity state $u(\mathbf{x},t)$, EuLaNet constructs its induced Lagrangian flow map $X(\mathbf{a},t)$ through $\dot{X}(\mathbf{a},t)=u(X(\mathbf{a},t),t)$, from which material transport and finite-time deformation are derived. The resulting representation couples the predicted state to the dynamical consequences it induces, providing a second consistency criterion beyond state-level agreement. We formalize this construction through an effective hypothesis space $\mathcal{H}_{L^2}\subseteq\mathcal{H}$ and define the conditions under which a consequence representation provides discriminative constraints on candidate solutions. EuLaNet is implemented as a model-independent representation module, separating the physical constraint from the downstream learning architecture. This construction provides an ante-hoc mechanism for physically interpretable constraint in scientific learning and offers a basis for developing and evaluating broader classes of Learning$^2$ architectures. The implementation is open-sourced to support the development and extension of the architecture across scientific domains.