Gradient Hermite–Birkhoff Interpolation

• Gradient Hermite–Birkhoff interpolation is a kernel-based technique that constructs interpolants using only first-derivative data in a Reproducing Kernel Hilbert Space.
• The method guarantees symplectic structure preservation in Hamiltonian dynamics by generating discrete-time symplectic updates that enhance long-term simulation accuracy.
• Utilizing greedy center selection and rigorous RKHS theory, the approach achieves algebraic error decay and computational efficiency for structure-preserving learning.

Gradient Hermite–Birkhoff interpolation is an operator-theoretic and kernel-based method for constructing interpolants of scalar functions whose gradients are prescribed at a collection of scattered points. This methodology has found particular utility within Hamiltonian dynamics, where the preservation of symplectic structure, energy, and phase-space invariants is essential for accurate long-term simulation and surrogate modeling. The approach generalizes classical Hermite interpolation by incorporating only first-derivative data, often in settings where function values are unavailable or otherwise irrelevant. Within the context of structure-preserving learning, gradient Hermite–Birkhoff interpolation provides a rigorous and computationally efficient means for constructing, training, and analyzing Hamiltonian surrogates, with guarantees on existence, uniqueness, error decay, and symplecticity.

1. Mathematical Formulation and Interpolation Problem

Gradient Hermite–Birkhoff interpolation—also referred to as first-derivative HB interpolation—is defined on a Reproducing Kernel Hilbert Space (RKHS) $H_k(\Omega)$, associated with a strictly positive definite kernel $k$. Given a set of mixed-argument points $$\{ \xi_j = (q_0^j, p_{\Delta T}^j) \}_{j=1}^M \subset \mathbb{R}^{2n}$$ and prescribed gradient values $y_j \in \mathbb{R}^{2n}$, the objective is to construct a function $\Phi \in H_k(\Omega)$ such that

$\nabla \Phi(\xi_j) = y_j, \quad j = 1, \dots, M.$

The solution is characterized by

$$\Phi(x) = \sum_{j=1}^M \sum_{\alpha = 1}^{2n} c_{j, \alpha}\, \partial^{(2)}_\alpha k(x, \xi_j),$$

where the coefficients $c_{j, \alpha}$ solve the linear system

$$G c = y, \qquad G_{(i, \beta), (j, \alpha)} = \partial^{(1)}_\beta \partial^{(2)}_\alpha k(\xi_i, \xi_j).$$

If the directional derivatives $\{ \lambda_{j, \alpha} \}$ are distinct and the kernel is strictly positive definite, a unique interpolant with minimal RKHS norm exists (Herkert et al., 26 Jan 2026).

This construction directly enforces the Hermite–Birkhoff interpolation conditions for gradients, with no requirement for interpolation of function values. The approach is flexible with respect to kernel choice (e.g., Gaussian, inverse multiquadric, Matérn), provided sufficient regularity.

2. Application to Symplectic Structure Preservation

When gradient Hermite–Birkhoff interpolation is applied to the scalar generating function of a symplectic map, the resulting interpolant yields a discrete-time symplectic update. For canonical coordinates $x = (q, p)$, and denoting $\Phi$ as the learned surrogate, the flow map is defined via the symplectic Euler integration

$$\begin{cases} p_{n+1} &= p_n - h\, \nabla_q \Phi(q_n, p_{n+1}), \ q_{n+1} &= q_n + h\, \nabla_p \Phi(q_n, p_{n+1}), \end{cases}$$

where $h = \Delta T$ is the macro-step. This map $\Psi_s$ is symplectic by construction: $[D\Psi_s(x)]^\top J_{2n} D\Psi_s(x) = J_{2n},$ with $J_{2n}$ the canonical Poisson matrix. The gradient data in the interpolation problem derive from exact or numerically generated flow ($x_0 \mapsto x_{\Delta T} = \Phi^{\Delta T}(x_0)$), leading to the identification

$$\nabla \Phi(q_0, p_{\Delta T}) = J_{2n}^\top \frac{x_{\Delta T} - x_0}{\Delta T}.$$

Thus the SKP (Symplectic Kernel Predictor) architecture uses gradient HB interpolation to guarantee that the learned discrete flow preserves the symplectic structure for arbitrary step sizes (Herkert et al., 26 Jan 2026, Rath et al., 2020).

3. Existence, Uniqueness, and Algorithmic Realization

For strictly positive definite kernels and linearly independent functionals, the gradient HB interpolation Gram matrix is non-singular, and the minimum-norm interpolant exists uniquely (Herkert et al., 26 Jan 2026). The computation involves forming the block Gram matrix of second derivatives, solving the resulting linear system for the coefficients $c_{j,\alpha}$, and then evaluating the interpolating function and its gradient as needed.

For large datasets (i.e., high $M$), the method admits greedy center selection based on VKOGA or $f$-greedy procedures. At each step, the next interpolation condition (directional derivative at a location) is chosen as that with the largest residual, reducing the residual in a greedy fashion. This leads to sparse representations and controlled computational complexity in high dimensions.

Algorithmic steps:

1. Assemble mixed-argument points $\xi_j$ and corresponding gradients $y_j$.
2. Construct Gram matrix $G$ using all second derivatives of the RKHS kernel.
3. Optionally, perform $f$-greedy selection to build sparse interpolation sets.
4. Solve $Gc = y$ for $c$.
5. Form interpolant $\Phi$ as a linear combination of kernel derivatives.

4. Convergence Behavior and Error Propagation

A key theoretical property is the algebraic decay rate of the interpolation error in the RKHS norm and its propagation to prediction error in trajectory space. Let $e_m = \Phi^\star - s_m$, where $s_m$ is the $m$-center interpolant. The block-wise error for the gradient satisfies, for all $m \geq 1$,

$$\min_{m+1 \leq i \leq 2m} \| \nabla e_i \|_{L^\infty(\Omega)} \leq \sqrt{n} m^{-1/2} \| e_{m+1} \|_{H_k(\Omega)} \left[ \prod_{i=m+1}^{2m} P_i \right]^{1/m},$$

where $P_i$ denotes the RKHS power-function for the interpolation set (Herkert et al., 26 Jan 2026). Empirically, $P_i$ also decays algebraically; as a consequence, the gradient HB error decreases almost algebraically in the number of centers. This error decay propagates to the one-step prediction error through bounds of the form

$$\| x_{\Delta T, i_m}(x_0) - \Phi^{\Delta T}(x_0) \|_2 \leq C \Delta T \sqrt{2n} m^{-1/2} \| e_{m+1} \|_{H_k(\Omega)} \left[ \prod_{i=m+1}^{2m} P_i \right]^{1/m},$$

under suitable solvability and regularity conditions on the exact flow.

5. Relation to Kernel Methods for Structure-Preserving Learning

Gradient Hermite–Birkhoff interpolation is deeply connected to the recent class of structure-preserving kernel methods for learning Hamiltonian dynamics. The SKP and related estimators can be cast as solutions to regularized least-squares problems involving loss functions of vector field gradients, with the differential representer theorem guaranteeing that the minimizer is an RKHS linear combination of kernel gradients (Hu et al., 2024, Smith et al., 2023, Smith et al., 2024).

For trajectory-based learning, the relation between function interpolation (as in standard Hermite interpolation), gradient-only interpolation (gradient Hermite–Birkhoff), and kernel regression is clarified: the minimum-norm solution with prescribed gradients at selected points is equivalent to the posterior mean estimator under a Gaussian process prior, when the GP is placed on the generating function or Hamiltonian itself (Rath et al., 2020, Hu et al., 2024).

The approach is compatible with further model order reduction, e.g., through symplectic SVD bases in high-dimensional discretized PDEs, with the structure-preserving nature inherited post-projection (Herkert et al., 26 Jan 2026).

6. Numerical Performance and Empirical Observations

Benchmark studies on canonical systems such as the pendulum, nonlinear spring–mass chains, and the semi-discrete wave equation have demonstrated algebraic convergence rates and qualitatively improved long-term trajectory accuracy over implicit midpoint baselines. For instance, in the pendulum ($n = 1$), greedy residuals decay to $10^{-6}$ with roughly 200 centers, and SKP one-step prediction errors are $10^{-7}$ to $10^{-5}$ compared to $10^{-3}$ to $10^{-1}$ for implicit midpoint methods at comparable step sizes. In higher dimensions, the error trend is confirmed, enabling accurate, stable integration with significantly fewer macro time steps (Herkert et al., 26 Jan 2026).

The method tracks training and validation residuals closely, shows bounded oscillatory prediction error (characteristic of symplecticity), and generalizes well to previously unseen trajectories. Overfitting is not observed when using greedy center selection, and the approach is robust to variation in data sampling.

7. Broader Impact, Limitations, and Future Directions

Gradient Hermite–Birkhoff interpolation underpins a family of structure-preserving surrogate models and integrators for Hamiltonian and symplectic systems, achieving data-driven energy and phase-space conservation in both low- and high-dimensional settings. The method relies on rigorous functional-analytic underpinnings, enables sparse and efficient algorithms, and bridges the gap between classical interpolation theory and modern kernel machine learning.

Open challenges include the further acceleration of linear solves for very large interpolation sets, extension to higher-order symplectic integrators via higher-derivative interpolation conditions, and generalization to noncanonical or non-Hamiltonian geometric structures. The potential for adaptive kernel selection and integration with learning-based hyperparameter optimization schemes remains a subject of ongoing research (Herkert et al., 26 Jan 2026, Rath et al., 2020).