Symplecticity‐Preserving Prediction of Parameter‐Dependent Hamiltonian Dynamics by Generalized Kernel Interpolation

We extend the kernel‐based symplectic predictor introduced by Herkert et al. (2026) to a parameter‐augmented setting in which the learned flow‐map surrogate depends not only on the state, but also on additional variables such as physical parameters and macro time‐step sizes. The method uses a product kernel ansatz on a parameter and macro step augmented domain and constructs the prediction through an implicit symplectic‐Euler‐type update. Hence, for every fixed admissible parameter and time‐step instance, the resulting large‐step predictor is symplectic by construction. The training problem is formulated as gradient Hermite–Birkhoff interpolation in a reproducing kernel Hilbert space. Efficient surrogates are obtained by greedy center selection. We show that the convergence analysis from the non‐augmented setting carries over to the product kernel framework and derive corresponding prediction error bounds. Numerical experiments for a pendulum with varying length and time‐step size and for a parameter‐dependent discretized wave equation illustrate the accuracy and structure‐preserving behavior of the proposed approach.