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Inference of Hamiltonian and Lie-Poisson structures from vector field samples

Jason E. Frank, Georg A. Gottwald

arXiv:2610.04350v1math.NAcs.NAmath.DSmath.SG

TL;DR

Estimating Hamiltonian dynamics from vector-field samples while also learning an unknown Poisson structure remains an open problem. The paper uses random-feature Hamiltonians and regression-based methods to learn Hamiltonians and structures across several Poisson classes. It reports accurate system identification and long-term prediction using the inferred structure.

  • Problem

    Estimating the Hamiltonian and Poisson structure simultaneously remains an open problem, despite the need to preserve Hamiltonian geometry in system identification.

    Source: §1 · pp. 2–3

  • Method

    The method represents the Hamiltonian with random features and jointly estimates it and unknown structure parameters using regression and constrained optimization.

    Source: §1 · p. 3

  • Results

    The approach infers the Hamiltonian structure needed for accurate system identification and long-term prediction.

    Source: §1 · p. 3, §6 · p. 26

  • Takeaways & Limitations

    Poincare sections provide a demanding long-time test because inaccuracies in the Hamiltonian or structure can distort or change invariant-tori topology.

    Source: §5.1.1 · p. 17

    The study assumes state and vector-field samples; noisy state-only measurements and partial observations are not considered.

    Source: §6 · p. 27

Abstract

from arXiv · show

We present a method for the data-driven inference of Hamiltonian and Lie-Poisson dynamical systems from vector-field samples, in which the Hamiltonian and the underlying Poisson tensor are estimated jointly rather than assuming the structure is known. The Hamiltonian is represented as a linear combination of random feature maps with fixed random internal weights, so its estimation is a linear regression problem for any given Poisson structure. We treat three cases of increasing generality: a known Poisson tensor, an unknown constant invertible (symplectic) structure matrix, and Lie-Poisson and affine Poisson structures, with structure matrix linear in the state. In the constant case, the Hamiltonian and structure matrix are recovered jointly from an eigenvalue problem that avoids the trivial zero solution due to the scaling symmetry of Hamiltonian systems. For Lie-Poisson and affine Poisson systems we alternate between an initial Hamiltonian estimate from a generalized eigenvalue problem for an approximate first integral, and a constrained nonlinear least-squares fit of the structure constants that enforces the Jacobi identity and, for affine structures, the cocycle condition, via continuation in a penalty parameter. Orthogonal matching pursuit selects a compact, well-conditioned subset of random features. Because the learned Hamiltonian is a sum of single-feature terms, the model admits an explicit Poisson/symplectic splitting integrator, which we use to validate the learned systems via long-time Poincare sections. We demonstrate the approach on the four- and five-dimensional Lorenz-86 model and a six-dimensional Kirchhoff rigid-body system, recovering Hamiltonians, structure constants and Casimir invariants to high accuracy, and showing the inferred Poincare sections converge to the correct invariant-tori topology as the number of random features increases.

1 Introduction

The paper addresses system identification for Hamiltonian systems, where estimating a vector field alone can fail to preserve its geometric structure. It proposes learning the Hamiltonian and, when unknown, the Poisson structure from vector-field samples.

1 Introduction

  • System identification for Hamiltonian dynamics is difficult because methods successful for dissipative systems may fail to preserve Hamiltonian geometry, limiting long-term integration.The introduction notes that such methods typically imply nonphysical energy drain.Source: §1 · p. 2
  • The open problem is to estimate the Hamiltonian and Poisson structure simultaneously, including for Lie-Poisson and affine Poisson systems.These systems have structure matrices linear in the state variables.Source: §1 · p. 2
  • The method represents the Hamiltonian with random features and uses regression to estimate it together with structure constants.A continuation strategy in a nonlinear least-squares solver enforces the Jacobi identity.Source: §1 · p. 3
  • The paper studies known Poisson tensors, unknown constant symplectic structures, and unknown Lie-Poisson or affine Poisson structures.Its numerical tests include four- and five-dimensional Lorenz-86 and a six-dimensional Kirchhoff rigid-body model.Source: §1 · p. 3

2 Hamiltonian dynamical systems and Lie-Poisson systems

Hamiltonian dynamics are generated by an energy function and a skew-symmetric structure, with Poisson systems allowing more general, potentially state-dependent structure tensors. Preserving these structures matters for long-time simulation and the associated geometric properties.

2 Hamiltonian dynamical systems and Lie-Poisson systems

  • Symplecticity constrains Hamiltonian dynamics, implying volume preservation and energy conservation, while supporting long-time preservation by symplectic numerical methods.The passage also connects symplecticity with robustness of integrable structures under small perturbations through KAM theory.Source: §2 · p. 4
  • A nondegenerate skew-symmetric symplectic matrix requires an even-dimensional state space and can be transformed from canonical coordinates to a generally noncanonical full matrix.The canonical coordinates partition the state into conjugate coordinates and momenta.Source: §2 · p. 4
  • A Poisson tensor is a local matrix function that must be skew-symmetric and satisfy the Jacobi identity.When degenerate, it may admit Casimir functions that are first integrals for any Hamiltonian.Source: §2 · pp. 4–5
  • A Lie-Poisson structure has a tensor linear in the state, while an affine Poisson structure adds a constant skew-symmetric tensor.Affine structures also require the cocycle condition.Source: §2 · p. 5

3 Random feature maps

The paper uses random-feature representations to infer Hamiltonian systems across known and unknown Poisson structures. It combines regression, eigenvalue methods, and constrained optimization, including an initialization from an approximate first integral for affine and Lie-Poisson systems.

3 Random feature maps

  • Random features keep internal weights fixed and estimate outer weights by explicit least-squares regression; hit-and-run sampling avoids saturated or nearly linear tanh responses.Orthogonal matching pursuit can reduce the feature pool.Source: §3 · pp. 6–7

3.1 Learning a Hamiltonian system with known Poisson structure

  • With a known Poisson tensor, the Hamiltonian coefficients are found by ridge-regularized regression, whose penalty also limits stiffness in the learned vector field.The implementation solves the overconstrained system using a singular value decomposition.Source: §3.1 · pp. 8–9

3.2 Learning a Hamiltonian system with unknown constant nondegenerate structure matrix

  • For an unknown constant symplectic matrix, joint estimation uses an eigenvalue problem and a normalization that avoids the trivial zero solution caused by Hamiltonian-system scaling symmetry.The learned structure is then used to reduce the feature pool before fitting the Hamiltonian coefficients.Source: §3.2 · pp. 9–10

3.3 Learning a Hamiltonian system with Lie-Poisson or affine Poisson structure

  • For affine and Lie-Poisson systems, the method alternates between fitting structure constants under Jacobi and cocycle constraints and solving for Hamiltonian coefficients.The structure tensor is linear in its coefficients, which are constrained by the Jacobi identity and, for affine structures, the cocycle condition.Source: §3.3 · p. 11

3.3.1 Initialisation step: Estimating a first integral from vector field data

  • Initialization estimates a nontrivial first integral as a random-feature function by a generalized eigenvalue problem, then uses it as the initial Hamiltonian estimate.The estimate may combine the Hamiltonian with Casimirs or Noether momenta; Noether momenta could degrade the initial guess.Source: §3.3.1 · pp. 11–12

3.3.2 Nonlinear least squares for structure constants

  • The structure-constant fit uses weak-constraint nonlinear least squares with continuation, gradually enforcing Jacobi and cocycle constraints as the penalty increases.The continuation addresses the reducible, stratified, nonsmooth constraint variety that obstructs gradient minimization with projection.Source: §3.3.2 · p. 13

4 Numerical integration of the inferred model via a Poisson splitting

The paper constructs explicit Poisson or symplectic splitting integrators by splitting the learned Hamiltonian into single-feature terms. The resulting schemes preserve the relevant structure, though splitting errors can require smaller steps for large feature counts.

4 Numerical integration of the inferred model via a Poisson splitting

  • Splitting methods provide explicitly computable alternatives to implicit Gauss–Legendre schemes for integrating inferred Hamiltonian or Lie-Poisson systems.The first-order splitting has cost comparable to evaluating the full vector field; the symmetric splitting is twice as expensive.Source: §4 · pp. 14–15
  • Splitting the learned Hamiltonian into single-feature terms yields compositions of exact feature flows that are symplectic for constant structures.The Lie–Trotter composition is first order, and its symmetric Strang composition is second order.Source: §4 · pp. 14–15
  • For large K, splitting errors may become significant and can require a smaller step size than integration of the true Hamiltonian.The scheme generalizes recursively to larger feature counts.Source: §4 · p. 15
  • For affine and Lie-Poisson systems, feature flows use a matrix exponential and yield first- or second-order Poisson integrators.Computing 2K matrix exponentials per time step may become costly.Source: §4 · p. 15

5 Numerical results

The numerical study tests inference on Lorenz-86 and Kirchhoff systems spanning symplectic, affine Poisson, and Lie-Poisson structures. The computations use noise-free vector-field samples and randomly sampled internal feature weights.

5 Numerical results

  • The study tests a four-dimensional symplectic and a five-dimensional affine-Poisson Lorenz-86 model, plus a six-dimensional Kirchhoff Lie-Poisson system.The Lorenz-86 model is considered in two versions.Source: §5 · p. 16
  • All computations assume noise-free vector-field data.Source: §5 · p. 16
  • Random feature-map internal weights and biases are sampled using the hit-and-run algorithm [28].Source: §5 · p. 16

5.1 The models

The models section introduces the Lorenz-86 and Kirchhoff test systems and defines the Poincaré section used to assess learned-model accuracy. The Lorenz-86 setup includes a four-dimensional canonical form with invariant tori and chaotic dynamics.

5.1 The models

  • The models section introduces the Poincaré section as the accuracy test for the learned systems.Source: §5.1 · p. 16

5.1.1 The Lorenz 1986 model

  • The five-dimensional Lorenz-86 system has an affine Poisson structure and a Casimir; restricting to a constant-Casimir level yields a canonical four-dimensional symplectic model.The section uses that four-dimensional representation.Source: §5.1.1 · pp. 16–17
  • The Lorenz-86 test section uses the (φ, x3) plane at x4 = 0 with ẋ4 > 0, and significant model errors can distort or change invariant-tori topology.The section is computed from 50 initial conditions over simulations to T = 2000; Figure 1 gives the exact-model reference section.Source: §5.1.1 · p. 17, Figure 1 · p. 18

5.1.2 The Kirchhoff equations

  • The Kirchhoff rigid-body model describes motion in an ideal fluid on R6 and is a Lie-Poisson system with two Casimir functions.Source: §5.1.2 · p. 18

5.2 Inferring the Hamiltonian for a given canonical structure

With the canonical symplectic structure known, the method fits the Hamiltonian from vector-field samples and uses OMP to reduce the random-feature pool. Increasing the feature pool improves the learned Poincaré sections toward the exact invariant-tori topology, while pool size and ridge regularization also affect fit accuracy and stiffness.

5.2 Inferring the Hamiltonian for a given canonical structure

  • For a known canonical structure, the method estimates the canonical Hamiltonian from vector-field data by minimizing the stated regression objective.Source: §5.2 · p. 18
  • Pool size Kpool and ridge parameter β indicate how to target vector-field accuracy and provide a rough indication of the learned field’s stiffness.The study tracks both vector-field error and the largest-magnitude learned coefficient, |α|∞.Source: §5.2 · p. 19
  • OMP selects a substantially reduced feature pool while retaining good approximation error.The reported table compares initial and selected pool sizes.Source: §5.2 · p. 19
  • As Kpool rises from 2000 to 16000, vector-field error falls from 0.0428 to 0.0021 and the Poincaré sections converge to the exact invariant-tori topology.The corresponding selected pool sizes are 424, 524, 635, and 763; the step size is fixed at 0.00125.Source: §5.2 · p. 19

5.3 Inferring the Hamiltonian and the symplectic structure

The section jointly infers the Hamiltonian and symplectic structure from transformed vector-field data. The learned structure is highly accurate, while Poincaré-section fidelity depends on the training data and is not fully converged in every setting.

5.3 Inferring the Hamiltonian and the symplectic structure

  • Figure 2 maps vector-field error and maximal learned weight across RFM pool size Kpool and ridge parameter β, showing accuracy and stiffness-related tuning information.The plotted isolines are best fits to learned-model results shown as points in vertical columns.Source: Figure 2 · p. 20
  • At Kpool = 12000 and M = 3600, the relative error in the inferred symplectic structure matrix J is 10^-9.The passage also notes that M must exceed Kpool/d because J is solved before orthogonal matching pursuit.Source: §5.3 · p. 20
  • With uniform random training samples and Kpool = 20000, the learned Poincaré sections remain slightly skewed and their topology is not fully converged.The comparison uses transformed coordinates and their inverse transformation to canonical coordinates.Source: §5.3 · pp. 19–20
  • When training samples are drawn from orbits covering the Poincaré section, that section is reproduced very well with Kpool = 12000.The reported integration step size is ∆t = 0.00125.Source: §5.3 · pp. 20–21

5.4 Inferring the Hamiltonian and an affine Poisson structure

The affine Poisson inference fits the L86 vector field and structure coefficients, then validates the learned model with Poincaré sections and conserved quantities. The learned sections are close to the exact sections, and its exact conserved quantities have small relative errors.

5.4 Inferring the Hamiltonian and an affine Poisson structure

  • The affine Poisson fit reaches relative vector-field error 0.00036 and determines structure coefficient C to relative error 2.2 × 10^-4.The fit uses 500 random features without OMP and 4000 training points.Source: §5.4 · p. 21
  • The learned Poincaré section is already very close to the exact section, despite using integration step size ∆t = 0.02 versus ∆t = 0.0025 for the true system.Both systems use a symmetric Poisson splitting integrator.Source: §5.4 · p. 22, Figure 6 · p. 23
  • The learned Casimir is slightly perturbed from the true Z, remaining quadratic in x but with small dependence on all components and an additional small linear term..Source: §5.4 · p. 22
  • Relative errors in the learned system’s exact conserved quantities are about 10^-7 for the Hamiltonian and 10^-12 for the Casimir, versus about 10^-3 for the true system’s conserved quantities.Figure 7 compares errors for the true-system invariants and the learned system’s exact invariants.Source: §5.4 · p. 23, Figure 7 · p. 24

5.5 Inferring an affine Poisson structure for a quadratic Hamiltonian

Assuming a quadratic Hamiltonian allows the affine L86 structure to be learned directly, and its isotropized structure constants are recovered to machine precision. The learned model’s Poincaré section is shown for 2,000 data points.

5.5 Inferring an affine Poisson structure for a quadratic Hamiltonian

  • A linear coordinate change makes a positive-definite quadratic Hamiltonian isotropic, reducing identification to learning the Poisson structure matrix.In the Lie-Poisson case, the linear term and translation are zero; the isotropic form removes feature maps and their coefficients from the loss.Source: §5.5 · p. 24
  • The isotropized affine structure constants were recovered to machine precision relative error for the quadratic-Hamiltonian L86 model.The experiment used 2,000 uniformly sampled vector-field data points.Source: §5.5 · p. 24
  • Figure 8 shows the learned L86 model’s Poincaré section using 2,000 data points and a stepsize of 0.01.The affine Poisson structure is inferred under the quadratic-Hamiltonian assumption.Source: Figure 8 · p. 25

5.6 Inferring the Hamiltonian and the Lie-Poisson structure

For the six-dimensional Kirchhoff model, the learned Lie algebra structure constants closely match the true constants, and the true and learned trajectories agree well. The learned system conserves its own Hamiltonian and Casimirs closely, although its Casimirs may differ from the true system’s.

5.6 Inferring the Hamiltonian and the Lie-Poisson structure

  • The learned Lie algebra structure constants had a maximum entrywise error of 0.000132.The fit used 6,000 phase-space samples and 500 random feature maps without OMP.Source: §5.6 · p. 25
  • The true and learned Kirchhoff trajectories show good agreement through time T = 300.Figure 9 compares Ω(t) and Γ(t), integrated with Poisson splitting at stepsize 0.01.Source: §5.6 · p. 25, Figure 9 · p. 26
  • The learned system’s Casimirs may differ significantly from the true system’s because multiple Casimirs are ambiguous.Figure 10 distinguishes conservation errors measured against the true model’s integrals from those measured against the learned model’s integrals.Source: §5.6 · p. 25
  • The learned system’s Casimirs are conserved to relative error below 10^-10 over the full time window.The observed drift grows at order t^1/2, consistent with random rounding errors; the splitting uses 1,000 matrix exponential computations per step.Source: §5.6 · p. 26

6 Discussion and outlook

The paper presents random-feature system identification for symplectic and affine Poisson Hamiltonian systems, learning the Hamiltonian and Poisson structure together. It reports recovery beyond dimension five, while noting assumptions about observed states and vector-field samples and proposing extensions to sparse libraries and coordinate transformations.

6 Discussion and outlook

  • The random-feature methodology identifies general symplectic and affine Poisson Hamiltonian systems, including Lie-Poisson systems, using bilevel optimization and gradual enforcement of Jacobi constraints.The paper reports successful recovery even when dimension exceeds five and Jacobi constraints outnumber unknowns.Source: §6 · p. 26
  • The method assumes access to the full state, including conjugate momenta in canonical coordinates, and to vector-field samples.Noisy state-only measurements require estimating the vector field; partial observations are not treated, though delay embedding may be used.Source: §6 · p. 27
  • A prescribed function library could replace random features and enable sparse identification of Lie-Poisson systems.Source: §6 · p. 27
  • Allowing a diffeomorphic coordinate transformation could test whether a system admits a Lie-Poisson structure, but makes the transformation coefficients enter the loss nonlinearly.Source: §6 · pp. 27–28
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