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Symplecticity-Preserving Prediction of Hamiltonian Dynamics by Generalized Kernel Interpolation

Robin Herkert, Tobias Ehring, Bernard Haasdonk

TL;DR

This work addresses long-time integration of Hamiltonian dynamics while preserving the canonical symplectic structure. It introduces a kernel-based surrogate that learns a scalar potential $s$ in an RKHS, whose gradient enters a discrete, symplectic-Euler update to predict the flow over a macro step $\Delta T$. The authors develop a gradient Hermite–Birkhoff interpolation framework, prove existence and convergence results (including a bound for gradient HB $f$-greedy), and integrate structure-preserving MOR to handle high-dimensional PDE discretizations. Across pendulum, nonlinear spring–mass chains, and discretized wave equations, the method achieves long-time accuracy improvements of two to three orders of magnitude over a baseline symplectic method, with strong generalization and scalable training thanks to sparse greedy selection.

Abstract

In this work, a kernel-based surrogate for integrating Hamiltonian dynamics that is symplectic by construction and tailored to large prediction horizons is proposed. The method learns a scalar potential whose gradient enters a symplectic-Euler update, yielding a discrete flow map that exactly preserves the canonical symplectic structure. Training is formulated as a gradient Hermite--Birkhoff interpolation problem in a reproducing kernel Hilbert space, providing a systematic framework for existence, uniqueness, and error control. Algorithmically, the symplectic kernel predictor is combined with structure-preserving model order reduction, enabling efficient treatment of high-dimensional discretized PDEs. Numerical tests for a pendulum, a nonlinear spring--mass chain, and a semi-discrete wave equation show nearly algebraic greedy convergence and long-time trajectory errors reduce by two to three orders of magnitude compared to an implicit midpoint baseline at the same macro time step.

Symplecticity-Preserving Prediction of Hamiltonian Dynamics by Generalized Kernel Interpolation

TL;DR

This work addresses long-time integration of Hamiltonian dynamics while preserving the canonical symplectic structure. It introduces a kernel-based surrogate that learns a scalar potential in an RKHS, whose gradient enters a discrete, symplectic-Euler update to predict the flow over a macro step . The authors develop a gradient Hermite–Birkhoff interpolation framework, prove existence and convergence results (including a bound for gradient HB -greedy), and integrate structure-preserving MOR to handle high-dimensional PDE discretizations. Across pendulum, nonlinear spring–mass chains, and discretized wave equations, the method achieves long-time accuracy improvements of two to three orders of magnitude over a baseline symplectic method, with strong generalization and scalable training thanks to sparse greedy selection.

Abstract

In this work, a kernel-based surrogate for integrating Hamiltonian dynamics that is symplectic by construction and tailored to large prediction horizons is proposed. The method learns a scalar potential whose gradient enters a symplectic-Euler update, yielding a discrete flow map that exactly preserves the canonical symplectic structure. Training is formulated as a gradient Hermite--Birkhoff interpolation problem in a reproducing kernel Hilbert space, providing a systematic framework for existence, uniqueness, and error control. Algorithmically, the symplectic kernel predictor is combined with structure-preserving model order reduction, enabling efficient treatment of high-dimensional discretized PDEs. Numerical tests for a pendulum, a nonlinear spring--mass chain, and a semi-discrete wave equation show nearly algebraic greedy convergence and long-time trajectory errors reduce by two to three orders of magnitude compared to an implicit midpoint baseline at the same macro time step.
Paper Structure (11 sections, 8 theorems, 202 equations, 7 figures)

This paper contains 11 sections, 8 theorems, 202 equations, 7 figures.

Table of Contents

  1. Introduction
  2. Background on Generalized Kernel Interpolation
  3. Convergence rates for gradient-Hermite-Birkhoff $f$-greedy
  4. On the Existence of the Target Function
  5. Numerical Experiments
  6. (i) maximum residual error versus number of centers.
  7. (ii) Relative error over time.
  8. Pendulum
  9. Nonlinear spring--mass chain with fixed ends
  10. Discretized wave equation
  11. Conclusion and Outlook

Key Result

Theorem 1

Let $u \in H_k(\Omega)$, $\Omega \subset \mathbb{R}^{n}$, and apply the $f$-greedy algorithm of the previous subsection with index set $\mathcal{J} := \{1,\dots,2n\}$. Then, for every $m \ge 1$, where the (derivative) power function is defined by

Figures (7)

  • Figure 1: Comparison of the training data for the explicit and implicit method with training data from the whole domain
  • Figure 2: Pendulum: (a) $f$-greedy convergence vs. centers; (b) relative error over time.
  • Figure 3: Pendulum (reduced training set): (a) $f$-greedy convergence vs. centers; (b) relative error over time.
  • Figure 4: Pendulum generalization test: comparison of one trajectory obtained from the kernel scheme ($\Delta T = 0.025$) with the reference solution.
  • Figure 5: Mass--spring chain: (a) $f$-greedy convergence vs. centers; (b) relative error over time.
  • ...and 2 more figures

Theorems & Definitions (19)

  • Theorem 1: First-order HB bound
  • proof
  • Definition 1: Type II generating function
  • Theorem 2: Convergence rate for the prediction error
  • proof
  • Theorem 3: Existence of a generating function
  • proof
  • Theorem 4: Uniform invertibility on compact forward-invariant sets
  • proof
  • Remark 1
  • ...and 9 more