Physical fields on meshes require a separation between topology and geometry: conservation laws are topological and should be exact, while geometry, material response, and anisotropic coupling must be learned from data. Existing neural surrogates often mix these roles inside unconstrained message passing. We introduce Riemannian Hodge Message…

Inverse problems for stiff parabolic partial differential equations (PDEs), such as the inverse heat conduction problem (IHCP), are severely ill-posed: the forward map rapidly damps high-frequency interior structure before it reaches the boundary. Soft-constrained physics-informed neural networks (PINNs), which embed the PDE as a residual…

Hamiltonian Neural Networks and related structure-preserving dynamics models encode conservation laws, but their scalar Hamiltonian parameterizations inherit the spectral bias of deep networks, limiting their ability to learn stiff multi-timescale systems with coupled fast and slow dynamics. We introduce the Frequency-Structured Hamiltonian…