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Model reduction of port-Hamiltonian systems via neural networks

Silke Glas, Alexander Heinlein, Harald Monsuur, Hongliang Mu

arXiv:2608.30788v1math.NA

TL;DR

State-dependent pH matrices can make reduced-model evaluation depend on the full-order dimension, increasing computational cost. The paper combines GMG projection with a structure-preserving neural network, achieving comparable accuracy and reported speed-ups in a numerical example.

  • Problem

    State-dependent interconnection and dissipation matrices can leave ROM evaluation dependent on the full-order dimension, increasing computational cost.

    Source: §1 · pp. 2–3, §3 · p. 5

  • Method

    The method first constructs a GMG-POD-ROM, then trains a neural network to approximate reduced interconnection and dissipation matrices while preserving pH structure.

    Source: §1 · p. 3, §2 · pp. 4–5, §5 · p. 10

  • Results

    2.67 × speed-up over GMG-POD-ROM and 53.38 × over FOM were achieved, with accuracy close to GMG-POD-ROM in the numerical example.

    Source: §4 · p. 9, §5 · p. 11

  • Takeaways & Limitations

    The GMG-POD-NN-ROM provides a structure-preserving reduced model with substantially faster evaluation in the reported nonlinear mass-spring-damper example.

    Source: §5 · pp. 10–11

    Combining GMG reduction with a nonlinear approximation map was not considered and is identified as future work.

    Source: §5 · p. 11

Abstract

from arXiv · show

In this paper, we consider structure-preserving model reduction of port-Hamiltonian (pH) systems which extend classical Hamiltonian systems with dissipation and an input-output port. These pH systems are often used in multi-physics systems, as the interconnection of one or more \pH systems results again in a pH system. If particularly the system matrices associated with the interconnection and/or dissipation of a pH system are state-dependent, then the evaluation of standard reduced-order models (ROMs) may depend on the dimension of the original full-order model, resulting in high computational costs. To circumvent these high costs, we propose to use structure-preserving neural networks. In particular, we perform two steps: (1) we use the generalized manifold Galerkin projection to project the pH system onto the reduced space; then (2) we train a neural network to learn the map from the reduced-order state to the reduced-order interconnection and dissipation system matrices. To ensure that the resulting ROM is again a pH system, the architecture of the neural network is chosen such that the skew-symmetry and positive semi-definiteness of the reduced-order systems matrices are maintained. In a numerical example, we consider a nonlinear mass-spring-damper system with state-dependent system matrices. The numerical results show that the proposed method achieves a significant computational speed-up compared to the original \ROM with comparable accuracy.

1 Introduction

Port-Hamiltonian systems model energy-based multi-physics dynamics while preserving interconnection and dissipation structure. The paper targets reduced models whose state-dependent matrices otherwise retain full-order computational costs.

  • Port-Hamiltonian systems extend Hamiltonian systems with dissipation and input-output ports, supporting multi-physics interconnection.Source: §1 · p. 2
  • The Hamiltonian represents internal energy, and the power balance makes pH systems passive by accounting for supplied and dissipated power.Source: §1 · p. 2
  • Projection-based model reduction lowers simulation cost for high-dimensional pH systems, but generic reduction need not preserve pH structure.Source: §1 · p. 2
  • Prior work includes structure-preserving MOR for linear and nonlinear pH systems, pH differential-algebraic equations, GMG projection, and neural-network learning of reduced systems.Source: §1 · pp. 2–3
  • This paper uses GMG projection followed by a structure-preserving neural network to learn nonlinear reduced interconnection and dissipation matrices.Source: §1 · p. 3

2 Projection-based MOR for pH systems and the Generalized manifold Galerkin reduction

The GMG framework constructs reduced pH systems by approximating the state in a low-dimensional space and enforcing a projected residual condition. A POD-based linear map satisfies the theorem’s conditions and yields the GMG-POD-ROM.

  • An approximation map 𝜑 from reduced coordinates to the full state is chosen with reduced dimension r much smaller than N.Source: §2 · p. 3
  • The reconstructed state evolves through the Jacobian D𝜑 of the approximation map via the chain rule.Source: §2 · p. 4
  • GMG defines a reduced coordinate map using W(ˇx)⊤x and derives the ROM by requiring the projected time-continuous residual to vanish.Source: §2 · p. 4
  • For linear approximation maps, the theorem guarantees a reduced pH system when the approximation space contains the port-matrix range and the stated assumption holds.Source: §2 · pp. 4–5
  • The POD construction satisfies the theorem’s conditions, producing the GMG-POD-ROM.Source: §2 · p. 5

3 Neural network-learned reduced-order pH systems

State-dependent reduced matrices can make standard ROM evaluation depend on the full-order dimension. The proposed neural network approximates them while preserving skew-symmetry and positive semi-definiteness.

  • For state-dependent problems, evaluating reduced interconnection and dissipation matrices still depends on the FOM dimension.Source: §3 · p. 5
  • The standard reduced evaluation requires inversion of an N × N matrix, while interpolation may struggle to preserve positive semi-definiteness of the dissipation matrix.Source: §3 · p. 5
  • A neural network maps reduced states to matrix representations whose constructed outputs define skew-symmetric interconnection and symmetric positive semi-definite dissipation matrices.Source: §3 · pp. 5–6
  • The resulting GMG-POD-NN-ROM approximates the reduced matrices while maintaining the pH structure.Source: §3 · p. 6
  • Training uses reduced snapshots obtained by projecting the full snapshot matrix with the pseudoinverse of the projection basis.Source: §3 · p. 6
  • Standardization is applied to triangular and Cholesky parameterizations rather than the matrices directly, avoiding destruction of pH structure.Source: §3 · p. 6
  • The loss combines approximation of the reduced matrix difference with approximation of the system dynamics.Source: §3 · pp. 6–7

4 Numerical example

The numerical example evaluates the proposed reduction method on a nonlinear mass-spring-damper system with state-dependent dissipation. Results compare model accuracy and online simulation time for GMG-POD-ROM and GMG-POD-NN-ROM against the full-order model.

System setup

  • The benchmark uses a nonlinear mass-spring-damper system with state-dependent dissipation and a full-order dimension of 400.The system has n=200 masses, unit masses and coefficients, and is simulated on (0,10] with 500 snapshots.Source: §4 · pp. 7–8

ROM construction

  • The ROM construction first computes a linear approximation map and then builds an extended snapshot dataset for neural-network training.The approximation map is φ: ˇx ↦→ Vˇx, with V computed from the snapshot-based construction.Source: §4 · p. 8

Influence of nonlinear damping

  • The nonlinear damping term materially affects the full-order output, so setting c2=0 yields insufficient results.The comparison uses FOMs with c2=1 and c2=0, where the latter neglects nonlinear damping.Source: §4 · p. 9, Figure 2 · p. 9

ROM accuracy

  • For reduced orders above 6, the maximal deviations in reduction and output errors are 6.783·10^-3 and 7.891·10^-3, respectively.Both errors decrease as the reduced order increases, and the neural-network ROM remains close to the GMG-POD-ROM.Source: §4 · p. 9, Figure 3 · p. 10

Online simulation time

  • The FOM takes 193.77 s, while GMG-POD-ROM and GMG-POD-NN-ROM achieve average online times of 9.69 s and 3.63 s.These correspond to speed-ups of 20.00× and 53.38×, respectively, relative to the FOM.Source: §4 · p. 10, Figure 4 · p. 10

5 Conclusion and outlook

The proposed two-step GMG-POD-NN-ROM approximates reduced interconnection and dissipation matrices with a neural network while preserving the pH structure. In the numerical example, it retains comparable accuracy and substantially reduces simulation time, while nonlinear approximation maps remain future work.

5 Conclusion and outlook

  • The method first constructs a GMG-POD-ROM, then uses a neural network to approximate the reduced interconnection and dissipation matrices.The neural-network design preserves the pH structure of the reduced model.Source: §5 · p. 10
  • The neural-network ROM preserves the pH structure through the architecture used for approximating the reduced matrices.Source: §5 · p. 10
  • The evaluation of the reduced-order Hamiltonian gradient still depends on the FOM dimension, although a structure-preserving DEIM method can address this dependence.Source: §5 · p. 10
  • The GMG-POD-NN-ROM performs closely to the GMG-POD-ROM in terms of accuracy in the numerical example.Source: §5 · p. 11
  • 53.38 × speed-up is achieved against the FOM, while the speed-up against the GMG-POD-ROM is 2.67 ×.Source: §5 · p. 11
  • Combining GMG reduction with a nonlinear approximation map was not considered and is identified as future work.Source: §5 · p. 11
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