Physics-Induced Koopman Learning for Mesh-Based Dynamics

Finite-dimensional Koopman learning seeks to represent nonlinear dynamics through linear evolution in learned observable coordinates, but a generic learned realization need not respect the physical structure of the target system. We characterize families of linear latent generators with finite interaction radius, consistency across isomorphic local environments, and orientation covariance, showing that each such family is uniquely represented by a shared matrix-valued kernel on local interaction types. When the set of local interaction types is finite, this yields a finite expansion in orientation-graded relation operators with uniquely determined coefficients. We instantiate this reduction as Koopman-MeshFT, combining local covariant encoders and decoders with a growth-controlled continuous-time generator. Across four public PDE benchmarks, Koopman-MeshFT attains the lowest mean rollout error among the evaluated Koopman baselines and improves several benchmark-specific physical diagnostics over them, while remaining competitive with FNO as a nonlinear neural-operator reference. The learned linear realization also enables non-iterative state estimation and forecasting from sparse observations. These results establish a physics-induced hypothesis class for finite-dimensional Koopman learning without assuming exact finite-dimensional closure.