Strong imposition of Dirichlet boundary velocities in structure-preserving discretizations of elastodynamics

What the math gives to ML

The paper gives a structure-preserving way to impose time-dependent Dirichlet velocities exactly without Lagrange multipliers: decompose the state into a dynamic component that vanishes on the constrained boundary and a prescribed lifting component. The transferable asset is not the finite-element assembly itself, but the resulting boundary-conditioned coordinate system, in which the network never predicts an invalid boundary value and the input enters as an explicit distributed port. This can be used to build elastodynamics neural operators or neural ODEs whose outputs satisfy boundary kinematics exactly while a Hamiltonian or power-balance penalty controls stability. The most practical first test is to replace unconstrained displacement and velocity prediction in a rollout model with a boundary-vanishing residual plus an analytically constructed lift.