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Learning Hamiltonians from Trajectory Data with Kernel Methods

Jianyu Hu, David Martin de Diego, Juan-Pablo Ortega, Miguel Vaquero

arXiv:2610.17835v1math.NAcs.NAmath.SG

TL;DR

At finite sampling steps, structure-preserving learning may identify a scheme-dependent Hamiltonian rather than the physical one. The paper develops analytic inverse targets and RKHS recovery bounds, then shows that multistride Richardson correction raises the bias order and experiments confirm the predicted scheme-dependent behavior.

  • Problem

    At finite sampling steps, structure-preserving learning may identify a scheme-dependent Hamiltonian rather than the physical Hamiltonian.

    Source: §1 · pp. 2–3

  • Method

    The method constructs analytic inverse Hamiltonian targets, estimates them with RKHS kernel regression, and combines stride-based estimators through Richardson extrapolation.

    Source: §1 · pp. 2, 7, §7 · pp. 39–40

  • Results

    O_P(N^-γ/(2γ+2)) is attained by both integrators under balanced sampling; experiments confirm predicted orders and Richardson gains until learning errors dominate.

    Source: §7 · pp. 40–41

  • Takeaways & Limitations

    Multistride correction can achieve the same physical recovery accuracy with a larger observation step, and midpoint reconstructions extend to several additional Hamiltonian systems.

    Source: §1 · p. 7, §7 · p. 41

    The trajectory-block bound does not exploit within-trajectory dependence, leaving potential refinement of its dependence on L for future work.

    Source: §7 · p. 41

Abstract

from arXiv · show

We study the recovery of a Hamiltonian from discrete trajectory observations using structure-preserving kernel methods. When exact-flow data are fitted through a symplectic one-step scheme, the fixed-step learning target is generally not the physical Hamiltonian itself, but a scheme-dependent inverse modified Hamiltonian. For the implicit midpoint and $p$-implicit symplectic Euler methods, we establish the existence of analytic inverse targets and show that their discrepancies from the physical Hamiltonian are second and first order in the observation step, respectively. Within an RKHS framework, we derive finite-sample reconstruction bounds for random trajectories, allowing dependence among observations from the same trajectory, and separate the error into estimation, approximation, and integrator-induced components. We further establish stability under noisy trajectory observations and develop a multistride Richardson correction that systematically raises the order of the integrator-induced error without requiring additional trajectories. Numerical experiments confirm the predicted scheme-dependent orders, the effect of Richardson correction, and the applicability of the method across several Hamiltonian systems.

1 Introduction

This paper studies Hamiltonian recovery from discrete trajectories while accounting for how the symplectic integrator shapes the learned target. At a fixed step, fitting exact-flow observations can identify a scheme-dependent Hamiltonian rather than the physical one.

1 Introduction

  • At finite step, fitting exact-flow pairs through a symplectic integrator can select a scheme-dependent Hamiltonian rather than the physical Hamiltonian.Structure preservation therefore does not by itself ensure target preservation.Source: §1 · p. 4
  • The analytic inverse targets differ from the physical Hamiltonian at second order for implicit midpoint and first order for symplectic Euler.Midpoint symmetry gives an even analytic family in the step, while symplectic Euler generally has a first-order correction.Source: §1 · p. 6
  • Finite-sample recovery bounds separate estimation, approximation, and integrator-induced errors while allowing dependence among observations within each trajectory.At fixed step, more trajectories can recover the scheme-dependent target but cannot remove its discrepancy from the physical Hamiltonian.Source: §1 · p. 6
  • Velocity noise is amplified at scale η^-1/2, while anchor noise enters at scale η^-1 in the pathwise stability bound.The resulting physical-Hamiltonian reconstruction bound combines these noise effects with estimation, approximation, and integrator-induced errors.Source: §1 · p. 7
  • Multistride Richardson extrapolation cancels the first Q−1 step-size corrections using stride estimators from the same trajectories, relaxing the required step-size refinement.The statistical recovery rate is retained without requiring independence across strides, and the same physical recovery accuracy can be achieved with a larger observation step.Source: §1 · p. 7
  • Experiments confirm the predicted midpoint and symplectic-Euler error orders and show Richardson gains until estimation and approximation errors dominate.The midpoint estimator also reconstructs the double pendulum, Frenkel–Kontorova, and strongly non-convex examples; independent trajectories provide statistical averaging while within-trajectory observations remain dependent.Source: §1 · pp. 7–8

2 Hamiltonian systems, symplectic integrators, and inverse modified Hamiltonians

This section introduces the midpoint and p-implicit symplectic Euler schemes and establishes analytic, scheme-dependent Hamiltonians whose one-step maps reproduce the exact flow at sufficiently small steps.

2.2 Inverse modified Hamiltonians

  • The analysis assumes analytic Hamiltonians and restricts implicit maps to the locally unique branch continuous at δ = 0.The inverse targets are normalized at a reference point, but remain identifiable only up to an additive constant.Source: §2 · p. 9, §2.2 · pp. 11–12

2.1 Symplectic one-step methods

  • The midpoint rule is second-order and time-symmetric, while p-implicit symplectic Euler is first-order and not time-symmetric.The midpoint target expansion contains only even powers of δ; the adjoint q-implicit Euler variant reverses the sign of the linear correction.Source: §2.1 · pp. 10–11, §2.2 · p. 12
  • The scheme-adapted anchor and discrete velocity are computed from endpoint pairs, with the anchor located where the integrator evaluates the Hamiltonian vector field.For symplectic Euler, the p-implicit variant uses (q, P), while its adjoint uses (Q, p).Source: §2.1 · p. 11

2.2 Inverse modified Hamiltonians

  • For sufficiently small steps, each selected scheme has an analytic inverse target whose one-step map reproduces the exact time-δ flow.The target depends on the integrator and differs from the physical Hamiltonian, up to an additive constant.Source: §2 · p. 8, §2.2 · p. 12
  • Midpoint produces a quadratic rescaling of the Hamiltonian, whereas symplectic Euler introduces a first-order cross-term.For the harmonic oscillator, the midpoint construction is singular at |ωδ| = π and the Euler mixed-point construction at |ωδ| = π/2.Source: §2.2 · p. 13

3 RKHS reconstruction from random trajectories

The section formulates trajectory-based RKHS recovery using scheme-adapted observations of an inverse modified Hamiltonian, then develops statistical, computational, and identifiability analyses that account for trajectory dependence and coverage.

3 RKHS reconstruction from random trajectories

  • The reconstruction problem infers an unknown Hamiltonian from fixed-step trajectory data while preserving Hamiltonian structure and compatibility with the selected symplectic discretization.The approach first reconstructs the fixed-step inverse target in an RKHS and relates it to the physical Hamiltonian as δ → 0.Source: §3 · p. 13

3.1 Random trajectories and scheme-adapted observations

  • At fixed δ, exact-flow trajectory pairs provide derivative observations of the scheme-dependent inverse target, not generally of the physical Hamiltonian.The estimator is constructed without requiring knowledge or evaluation of the exact flow map.Source: §3.1 · pp. 13–14, §3 · p. 13
  • Observations within a trajectory may be dependent, so the finite-sample analysis treats each trajectory as an independent block rather than each anchor.The trajectories are independently sampled, while their states are generally dependent.Source: §3.1 · pp. 13–14

3.2 RKHS formulation and regularized problems

  • The inverse targets belong to the RKHS and differ from the physical Hamiltonian by O(|δ|^rI), with rMP = 2 and rSE = 1.The RKHS is chosen so derivative evaluations are continuous, supporting the derivative observations used in regression.Source: §3.2 · pp. 14–15

3.3 Operator and kernel representations

  • The regularized empirical and population problems have unique solutions, and the empirical estimator also admits a finite-dimensional differential-kernel representation.The operator formulation supports statistical analysis, while the kernel representation yields a finite-dimensional linear system for computation.Source: §3.3 · pp. 15–16

3.4 Identifiability and trajectory coverage

  • For fixed L, reducing δ lowers the O(δ^rI) target bias but shortens physical observation time, limiting how much trajectory information departs from the initial design.Full support is sufficient for identifiability up to an additive constant, but the positive spectrum of B_I determines the strength of representation of observable directions.Source: §3.4 · pp. 17–18

4 Error bounds and stability

The section bounds reconstruction of the fixed-step inverse modified Hamiltonian, separating statistical and approximation errors from integrator bias. It also establishes pathwise stability to noisy observations and balances these error sources to recover the physical Hamiltonian.

4 Error bounds and stability

  • The reconstruction error separates into estimation, approximation, and integrator-induced terms, with the last scaling as O(δ^rI), where rMP = 2 and rSE = 1.The integrator term reflects fitting the fixed-step inverse target rather than the physical Hamiltonian.Source: §4 · p. 19

4.1 Estimation and approximation errors

  • Independent trajectories are treated as observation blocks, so concentration bounds do not require independence among anchors within a trajectory.The block-concentration result controls fluctuations of the empirical trajectory-anchor operator.Source: §4.1 · pp. 19–20
  • With a source condition, the approximation error vanishes as η → 0 at rate η^γ, while the source norm sets its constant.The source condition places the target in the identifiable subspace and acts as a quantitative regularity assumption.Source: §4.1 · pp. 20–21

4 Error bounds and stability

  • Theorem 4.4 bounds reconstruction error with high probability by combining finite-sample estimation, regularized approximation, and integrator-induced error under its stated assumptions.The fixed-step target must satisfy the source condition; trajectory observations may be dependent within each trajectory.Source: §4.1 · p. 19, §4.2 · p. 21

4.2 Total error and consistency

  • At fixed step size, an error floor of order δ^rI remains once estimation and approximation errors are small, and more data or less regularization cannot remove it.This floor comes from the discrepancy between the inverse modified target and physical Hamiltonian.Source: §4.2 · p. 21
  • The implicit midpoint rule and symplectic Euler attain the same balanced reconstruction rate, while midpoint’s second-order inverse-target accuracy permits slower observation-step refinement than symplectic Euler.The required step-size refinements depend on the order of the integrator-induced error.Source: §4.2 · p. 22

4.3 Pathwise robustness under additive observation noise

  • Observation noise perturbs both regression outputs and inputs; velocity noise is amplified by η^−1/2, anchor noise by η^−1, and endpoint differencing adds δ^−1 sensitivity.Thus reducing the step lowers integrator error but may amplify observation noise.Source: §4.3 · pp. 23, 25

5 Richardson extrapolation and integrator-error reduction

Multistride Richardson combinations cancel leading corrections in the inverse modified Hamiltonian targets using stride-separated pairs from the same trajectories. They raise the integrator-bias order while trading off against amplification of stride-wise learning errors.

5.1 Common multistride data and estimators

  • The same uniformly sampled trajectories supply stride-separated pairs, whose exact-flow intervals yield regression targets at effective step sδ.Stride values are integer-valued to reuse the observed trajectories; Richardson itself applies to arbitrary step sizes.Source: §5 · p. 26, §5.1 · p. 27

5 Richardson extrapolation and integrator-error reduction

  • With Q strides, Richardson cancellation raises deterministic bias from O(δ^2) to O(δ^2Q) for implicit midpoint and from O(δ) to O(δ^Q) for symplectic Euler.The first Q − 1 inverse-modified corrections cancel exactly.Source: §5 · p. 26, §5.2 · p. 29
  • Separately trained stride estimators are combined by linear post-processing, without extra trajectories or higher-order numerical integrators.The extrapolation applies to learned inverse modified Hamiltonians, not to numerical trajectory approximations.Source: §5 · p. 26, §5.2 · p. 28

5.1 Common multistride data and estimators

  • Each stride estimator fits its own inverse modified target, and stride-balanced regularization gives all strides the same effective regularization parameter.The stride-s problem is the base-step regression with δ replaced by sδ.Source: §5.1 · pp. 27–28

5.2 Richardson extrapolation of the stride estimators

  • Richardson raises integrator-error order but amplifies stride-wise estimation and approximation errors by at most its amplification factor, without requiring independence across strides.The stride estimators reuse the same observations and are statistically dependent.Source: §5.2 · p. 29, §5.3 · p. 31

5.3 Richardson reconstruction error

  • Higher extrapolation levels are most useful when integrator error is comparable to or larger than learning error; once learning errors dominate, amplification can make them counterproductive.Increasing Q reduces integrator error from O(δ^rI) to O(δ^rIQ) but does not improve the statistical rate.Source: §5.3 · p. 31
  • Richardson extrapolation preserves the statistical reconstruction rate while allowing slower observation-step refinement to retain it.The required scaling changes from δN ≍ N^−γ/[rI(2γ+2)] to δN ≍ N^−γ/[rIQ(2γ+2)].Source: §5.3 · p. 31

6 Numerical experiments

The experiments test error decomposition and scheme-dependent recovery on the Hénon–Heiles system, then examine Richardson correction, observation noise, trajectory dependence, and reconstruction on three additional systems.

6.1 Experimental setting

  • The experiments use Matérn-5/2 kernels and compare midpoint and symplectic Euler observations constructed from exact-flow endpoint pairs.The Hénon–Heiles benchmark evaluates reconstruction on a fixed-momentum slice; the reported reconstruction error accounts for the Hamiltonian’s additive-constant ambiguity.Source: §6 · p. 31, §6.1 · p. 32

6.2 Fixed-step recovery and integrator-induced error

  • The fitted error slopes are approximately 2.03 for implicit midpoint and 1.01 for symplectic Euler, matching their predicted second- and first-order behavior.At sufficiently small estimation and approximation errors, integrator-induced error controls reconstruction error.Source: §6.2 · pp. 32–33, Figure 2 · p. 33

6.3 Multistride Richardson correction and noise sensitivity

  • Richardson correction reduces integrator-induced error from O(δ^2) to O(δ^4) and O(δ^6) for midpoint, and from O(δ) to O(δ^2) and O(δ^3) for symplectic Euler.Corrected midpoint estimators reach an estimation–approximation floor at small step sizes, while symplectic Euler benefits substantially over a wider range.Source: §6.3 · p. 34, Figure 4 · p. 35
  • At low noise, Richardson correction reduces integrator-induced error, but higher extrapolation orders become increasingly sensitive to noisy trajectory observations.For symplectic Euler, correction remains beneficial across a wider low-noise regime than for midpoint.Source: §6.3 · p. 35, Figure 5 · p. 36

6.4 Trajectory dependence

  • With a fixed budget of 400 observations, midpoint error increases as observations shift from more independent trajectories to longer, dependent trajectories.The allocation effect is less visible for symplectic Euler because its first-order integrator-induced error dominates.Source: §6.4 · p. 36, Figure 6 · p. 37

6.5 Additional Hamiltonian systems

  • Across three additional systems, midpoint reconstruction captures nonlinear double-pendulum geometry, periodic coupling structure, and strongly non-convex energy features.After additive-constant alignment, mismatch is small across most of the highly non-convex system’s interior, with larger errors near the boundary.Source: §6.5 · pp. 37–38, Figure 7 · p. 38, Figure 8 · p. 38

7 Conclusions and future work

The conclusions identify the scheme-dependent inverse modified Hamiltonian as the fixed-step target, quantify its discrepancy from the physical Hamiltonian, and summarize recovery bounds, noise effects, and Richardson correction. Experiments support the predicted orders and show the method’s applicability across several Hamiltonian systems.

7 Conclusions and future work

  • The inverse modified Hamiltonian is the exact fixed-step target, with discrepancy O(δ^2) for midpoint and O(δ) for symplectic Euler.The midpoint family is even in δ, while symplectic Euler generally has a nonzero first-order correction.Source: §7 · p. 39
  • At fixed δ, increasing N drives the estimator toward the scheme-dependent target, not H, so it cannot remove integrator-induced error.The reconstruction error separates into estimation, approximation, and integrator-induced contributions.Source: §7 · pp. 39–40
  • With common source exponent γ and uniformly bounded source norms, both integrators attain rate N^(-γ/(2γ+2)); integrator order changes the required step-size refinement.When the source norm may grow as δ^(-β), the rate becomes N^(-rIγ/[2(rI(γ+1)+β)]).Source: §7 · p. 40
  • Velocity noise is amplified at scale η^(-1/2) and anchor noise at η^(-1), so reducing δ can lower integrator error while amplifying observation noise.The noisy-data bound combines with the clean-data estimate to separate observation noise from estimation, approximation, and integrator-induced error.Source: §7 · p. 40
  • Multistride Richardson correction cancels the first Q−1 inverse-target corrections, reducing target bias while retaining the statistical rate up to amplification factor AI_Q.For fixed Q, it relaxes the step-size refinement; increasing Q is most useful while integrator error is comparable to or larger than learning error.Source: §7 · p. 40
  • Experiments show the predicted midpoint and symplectic Euler orders, Richardson improvement until other errors dominate, and accurate midpoint reconstructions on three additional systems.The additional systems are the double pendulum, Frenkel–Kontorova, and strongly non-convex examples.Source: §7 · p. 41
  • The trajectory-block bound does not exploit within-trajectory dependence, and balanced rates require unknown γ and uniform source bounds; higher-order schemes and data-driven parameter choices remain open directions.The authors also identify noise-law errors-in-variables analysis and finite-regularity extensions as future work.Source: §7 · p. 41

A Backward error analysis and inverse modified Hamiltonians

The appendix constructs exact inverse modified Hamiltonians for implicit midpoint and p-implicit symplectic Euler by locally inverting their respective exact-flow maps. Both constructions yield analytic Hamiltonians, with midpoint bias quadratic and even in δ, while symplectic Euler bias is generally first order.

Appendix overview

  • The midpoint construction locally inverts the exact-flow midpoint map, while the p-implicit symplectic Euler construction locally inverts the exact-flow mixed-endpoint map.Both constructions proceed directly from the exact flow rather than a general B-series construction.Source: A Backward error analysis and inverse… · p. 45
  • Symplecticity makes the one-form forced by the exact chord closed, yielding a genuine jointly analytic Hamiltonian on a fixed neighborhood.The resulting coefficient bounds provide uniform smoothness estimates and, under the stated native-space assumptions, bounds in HK.Source: A Backward error analysis and inverse… · p. 45
  • The midpoint target bias is quadratic and even in δ, whereas the symplectic Euler bias is generally first order with a full power-series expansion.Source: A Backward error analysis and inverse… · p. 45

A.1.1 Exact midpoint-map construction

The midpoint construction locally inverts the exact-flow midpoint map to obtain a jointly analytic Hamiltonian whose midpoint step reproduces the exact flow. Symmetry makes the normalized target even in δ, so its odd Taylor coefficients vanish.

A.1.1 Exact midpoint-map construction

  • The exact-flow chord defines a Hamiltonian vector field because symplecticity makes its associated one-form closed and exact on the convex tube.The normalized potential yields the inverse target Hamiltonian.Source: §A.1.1 · pp. 45, 47
  • A jointly holomorphic local inverse of the exact-flow midpoint map exists on a fixed tube for sufficiently small δ.The inverse identities hold on the tube and on the original compact set.Source: §A.1.1 · p. 46
  • The constructed Hamiltonian is jointly holomorphic and reproduces the exact flow through a midpoint step on the locally unique branch.The branch is locally unique after reducing the step-size bound.Source: §A.1.1 · pp. 47–48
  • The normalized target is even in δ, so its convergent Taylor expansion has no odd coefficients.The expansion converges together with every fixed number of spatial derivatives on compact subsets of the tube.Source: §A.1.1 · pp. 46, 48

A.1.2 Quadratic target bias

For midpoint, the inverse modified Hamiltonian is a genuine analytic target obtained by reversing the usual forward modified-Hamiltonian construction. Its normalized expansion is even in δ, giving a quadratic target bias.

A.1.2 Quadratic target bias

  • The normalized midpoint target is even in δ, so its odd Taylor coefficients vanish and its discrepancy from H is quadratic in δ.Source: §A.1.2 · p. 48
  • The midpoint inverse construction seeks a Hamiltonian whose midpoint step reproduces the given exact flow, rather than a Hamiltonian whose exact flow represents a numerical step.The exact-flow midpoint coordinate makes the inverse problem pointwise, and the analytic inverse yields a genuine analytic family.Source: §A.1.2 · p. 49
  • The convergent series of the analytic inverse target is its Taylor series in δ.Source: §A.1.2 · p. 49

A.2.1 Exact mixed-endpoint construction

The p-implicit symplectic Euler construction locally inverts an exact-flow mixed-endpoint map to define an analytic inverse target. Symplecticity makes the forced one-form closed, and the resulting Hamiltonian reproduces the exact flow through the scheme.

A.2.1 Exact mixed-endpoint construction

  • The mixed map has a jointly holomorphic local inverse on a fixed tube for sufficiently small δ, defining the inverse modified vector field.The construction uses the mixed endpoint coordinates (q, P).Source: §A.2.1 · p. 49
  • The normalized target has a convergent Taylor expansion, but its family is generally not even in δ.The series converges with every fixed number of spatial derivatives on compact subsets.Source: §A.2.1 · p. 50
  • Symplecticity makes the one-form forced by the exact-flow chord closed, and its normalized potential defines the analytic Hamiltonian target.The closedness identity extends to δ = 0, where the one-form is dH.Source: §A.2.1 · p. 51
  • The constructed Hamiltonian reproduces the exact flow through the p-implicit symplectic Euler step on the locally unique branch.Source: §A.2.1 · p. 51

A.2.2 First-order target bias

Corollary A.6 gives coefficient, native-space, and target-bias bounds under Assumption 2.4. The section distinguishes the inverse modified Hamiltonian from the classical forward construction and records the analytic tools and symbols used.

A.2.2 First-order target bias

  • Corollary A.6 establishes coefficient, native-space, and target-bias bounds under Assumption 2.4.Source: §A.2.2 · p. 52
  • The gradient order is generically sharp when ∇⟨Hq, Hp⟩ is not identically zero on K.Source: §A.2.2 · p. 52
  • Unlike the classical forward construction, the inverse construction uses a given exact flow to determine a Hamiltonian whose symplectic Euler step reproduces it.Its analytic family is an exact local object, and its convergent inverse modified series is its Taylor series in δ.Source: §A.2.2 · p. 52

B Complex-analytic tools

  • Appendix B collects classical complex-analysis results used in the backward error analysis, including holomorphic dependence of exact flows near time zero on compact sets.Source: B Complex-analytic tools · pp. 52–53

C Glossary of symbols

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