Structure-Preserving Model Order Reduction

• Structure-preserving model order reduction is a set of techniques that generate reduced-order models while exactly retaining structural properties such as energy conservation and passivity.
• It employs projection, interpolation, and optimization methods to maintain key forms like Hamiltonian, port-Hamiltonian, second-order, and skew-gradient in dynamical systems.
• Recent advancements achieve high accuracy and significant computational speed-ups in applications ranging from mechanical systems and power networks to complex multiphysics models.

Structure-preserving model order reduction (SP-MOR) comprises techniques that construct reduced-order models (ROMs) of dynamical systems while exactly retaining their underlying structural properties—such as Hamiltonian, port-Hamiltonian, second-order, passivity, or skew-gradient structure. These properties encode conservation laws, stability, and physical coupling present in the original system and are crucial for producing ROMs that remain physically meaningful and numerically robust. Recent developments in this field span projection-based algorithms, interpolation frameworks, balancing strategies, and data-driven or neural approaches, and address systems ranging from classical mechanical models to power networks, nonlinear DAEs, and parametric Hamiltonian PDEs.

1. Geometric and Algebraic Structure: Motivation and Foundations

Many high-dimensional dynamical systems are equipped with intrinsic geometric or algebraic structures. In mechanical modeling, these often arise as symplectic (Hamiltonian), port-Hamiltonian, weakly-damped second-order, or descriptor (DAE) forms, each encoding fundamental constraints and invariants:

• Hamiltonian systems: The ODE $\dot{x} = J \nabla H(x)$ with $J^T = -J$, $H(x)$ the energy, preserves a symplectic structure and energy conservation law. Non-canonical generalizations (Poisson/Hamiltonian) appear in fluid, plasma, and wave dynamics (Hesthaven et al., 2021).
• Port-Hamiltonian systems (pH): The form $\dot{x} = (J-R)Qx + Bu$, with $J=-J^T$, $R=R^T \ge 0$, $Q=Q^T>0$, $B$ the port matrix, $y = B^T Q x$ for output, guarantees passivity and physical interconnectivity (Mamunuzzaman et al., 2022). Generalizations include DAEs with (hidden) algebraic constraints (Moser et al., 2022).
• Second-order mechanical systems: $M \ddot{q} + D \dot{q} + K q = Bu$, $y = C_p q + C_v \dot{q}$ retain symmetric, positive-definite mass/stiffness, and dissipation (Beddig et al., 2020, Aumann et al., 2022).
• Skew-gradient systems: $\dot{y} = S(y) \nabla H(y)$ with skew-symmetry $S^T = -S$, $H$ a first integral (not necessarily quadratic) (Miyatake, 2018).
• Structure in DAEs: Engineering networks, coupled multiphysics, and power grids yield nonlinear DAEs where constraints (e.g., Kirchhoff laws) are encoded in singular $E$ matrices and must be retained in reduced models (Nadeem et al., 2024).

Preserving these structures in ROMs ensures the inheritance of invariants, stability, and correct dynamical coupling.

2. Projection-Based Structure-Preserving Reduction

Classical model reduction projects the original high-dimensional dynamics onto a lower-dimensional subspace, constructed via Proper Orthogonal Decomposition (POD), Krylov subspaces, or snapshot-based techniques. For SP-MOR, the construction of the projection spaces and the form of projection must be modified to enforce structure preservation:

• Symplectic projection for Hamiltonian systems: Construct a basis $V$ with $V^T J V = J_r$. The reduced dynamics $\dot{z} = J_r \nabla_z H_r(z)$, $H_r(z) = H(Vz)$, inherits symplecticity and exact energy conservation (Hesthaven et al., 2021, Gong et al., 2016, Herkert et al., 2023). Ortho-symplectic bases can be built via cotangent lifts, complex SVD, or strong greedy algorithms with exponential convergence when Kolmogorov width decays rapidly.
• Structure-preserving Galerkin for skew-gradient/Poisson systems: Use $S_r(z) = V^T S(Vz) V$, ensuring the reduced system $\dot{z} = S_r(z) \nabla_z H_r(z)$ exactly preserves the first integral $H_r(z)$ (Miyatake, 2018).
• Port-Hamiltonian projection via symplectic subspaces: Project onto $Q$ with $Q^T J Q = J_{2k}$; the ROM $\dot{z} = (\hat{J} - \hat{R})\hat{H}z + \hat{B}u$, with projected structure matrices, remains port-Hamiltonian and passive (Mamunuzzaman et al., 2022). For systems with dissipation, block-diagonal projectors ensure proper treatment of energy exchange.
• Second-order mechanical/Vibro-acoustic models: Petrov-Galerkin projection with $V$ and (possibly distinct) $W$ yields lower-dimensional $M_r$, $D_r$, $K_r$ matrices; symmetry and positive-definiteness are maintained when $W=V$ or via SVD-based balancing. The method encompasses modal truncation, balanced truncation, and rational interpolation (Beddig et al., 2020, Aumann et al., 2022).
• DAE structure preservation: Partitioning and block-diagonal projections retain the singularity pattern of the $E$ matrix, ensuring constraints are preserved. ROMs remain DAEs, reducing both dynamic and algebraic variables without ODE conversion or linearization (Nadeem et al., 2024).

Algorithms are tailored to guarantee structural invariants by construction, using suitable subspace, congruence, or Petrov–Galerkin projections.

3. Structure-Preserving Interpolation and Balancing Methods

Beyond projection, structure can be preserved by imposing interpolation (moment matching) and balancing conditions compatible with the original system's structure:

• Interpolation-based SP-MOR: For pH systems, symplectic subspaces constructed from resolvent evaluations at prescribed frequencies guarantee that the reduced transfer function matches the original at those points, while retaining the pH property and passivity (Mamunuzzaman et al., 2022). In second-order systems, structure-preserving two-sided or one-sided projection achieves exact moment/interpolation, retaining the affine, frequency-dependent material laws (Aumann et al., 2022, Benner et al., 2020).
• Balanced truncation: Gramians (controllability/observability) are computed with respect to the physical structure (second-order, pH, delay, or stochastic structure). The balancing transformation is performed to preserve symmetry, positive definiteness, and—in delay/stochastic cases—the block structure of system matrices (Becker et al., 2020). Error bounds of the form $$\lVert H-H_r\rVert_\infty \leq 2\sum_{i=r+1}^n \sigma_i$$ (with $\sigma_i$ the (generalized) Hankel singular values) are retained.

Specialized variants exist for frequency-limited, passivity-preserving (positive real), and parametric or bilinear systems (Breiten et al., 2021, Benner et al., 2020).

4. Optimization-Based and Data-Driven Structure-Preserving MOR

Recent advances deploy parameter optimization and data-driven methodologies for constructing structured ROMs:

• Parameter optimization frameworks (SOBMOR): ROM matrices are parameterized to encode structural constraints (symmetry, passivity, etc.) directly (e.g., $J = -J^T$, $R = R^T\geq 0$ for pH), and the ROM is optimized to minimize an objective functional (e.g., sample-wise $\mathcal{H}_\infty$ error) with respect to the parameters. Levelled least-squares surrogates, adaptive sampling, and differentiable objectives allow gradient-based solvers to enforce high-fidelity model reduction while exactly retaining structure (Schwerdtner et al., 2020, Schwerdtner et al., 2021, Moser et al., 2022). For index-two pH-DAEs, a minimal parameterization ensures all problem-specific redundancy is eliminated (Moser et al., 2022).
• Neural and dictionary-based adaptive approaches: Neural autoencoders compress very high-dimensional Hamiltonian or port-Hamiltonian systems into low-dimensional latent representations. The structure is preserved by matching the energy functional and constructing latent-space equations that retain symplectic/port-Hamiltonian form via decoder Jacobian transformations—enabling stable model-based control and learning (Lepri et al., 2023). Dictionary-based and rank-adaptive methods dynamically update local symplectic (or other structure-preserving) bases, drastically improving efficiency for transport-dominated or parametric systems (Herkert et al., 2023, Hesthaven et al., 2020).
• Nonlinear ansatzes and nonlinear Galerkin: For problems where linear projection spaces are inadequate, nonlinear ansatzes (manifold Galerkin with snapshot-based modes or factorized approximations) combined with weighted Petrov–Galerkin projection extend structure preservation to model classes with pronounced nonlinear or advective behavior (Schulze, 2023).

Optimization-based SP-MOR provides accuracy competitive with or exceeding non-structured methods while ensuring that physical and geometric constraints are never violated.

5. Extensions: Nonlinearities, Descriptor Systems, and Stochastic/Delay Systems

• Nonlinear systems: Extended to nonlinear port-Hamiltonian and Poisson/Skew-gradient systems using DEIM and tensorial hyper-reduction for nonlinearities, along with energy-preserving time integrators (AVF, Kahan's method, midpoint) (Karasözen et al., 2019, Gong et al., 2016, Miyatake, 2018). Structure-preserving reduction of DAEs in power networks enables simultaneous reduction of dynamic and algebraic components (Nadeem et al., 2024).
• Parametric and time-delay systems: Structure-preserving interpolation and balancing generalize to parametric bilinear systems, delay-differential equations, and stochastic systems with multiplicative noise. Moment-matching with respect to frequency, time, and parameter derivatives is achieved while strictly maintaining structure (Benner et al., 2020, Becker et al., 2020).
• Descriptor systems with hidden constraints: Algorithmic frameworks address index-one and index-two pH-DAEs, handling algebraic constraints in the reduction and supplying compact parametrizations amenable to optimization (Schwerdtner et al., 2022, Moser et al., 2022).

Structure preservation is leveraged to sustain model stability, invariants (e.g., Casimirs, energy, dissipation), and passivity over long time integrations and in parametric studies.

6. Practical Aspects, Benchmarks, and Comparative Results

Extensive numerical experiments demonstrate that SP-MOR methods:

• Achieve $\mathcal{O}(10^{-5})$–$\mathcal{O}(10^{-9})$ $\mathcal{H}_\infty$ or energy conservation accuracy, often surpassing classical (unstructured) projection methods by several orders of magnitude (Schwerdtner et al., 2020, Hesthaven et al., 2021, Breiten et al., 2021).
• Permit large-scale reduction (e.g., $n=10^5$ or $n=2000$ buses in power systems) with speed-ups of $10$–$100\times$, and with preservation of algebraic, physical and geometric structures (Lepri et al., 2023, Nadeem et al., 2024).
• Retain stability, passivity, and conservation laws in all tested settings—such as shallow water PDEs, mass-spring-damper networks, power networks, and vibrating structures (Karasözen et al., 2019, Beddig et al., 2020, Aumann et al., 2022).
• For parametric and transport-dominated problems, dictionary-based or rank-adaptive SP-MOR methods produce adaptive local bases, drastically reducing projection dimensions and computational costs (Herkert et al., 2023, Hesthaven et al., 2020).

Comparisons between SOBMOR, pH-balanced truncation, and unstructured IRKA/BT show that optimization-based SP-MOR can attain or approach the minimal model reduction error while always preserving the desired structure (Schwerdtner et al., 2020, Schwerdtner et al., 2021).

7. Outstanding Challenges and Research Directions

Despite substantial progress, several open directions remain in SP-MOR:

• Extension of structure-preserving IRKA and interpolation frameworks to generalized (affine, frequency-dependent) operators and fully nonlinear settings (Aumann et al., 2022).
• A priori error estimators for energy/Casimir errors and for frequency- and parameter-domain interpolation in nonlinear and DAEs.
• Integration of physics-informed neural networks, variational and symplectic autoencoders to enable structure-preserving reduction of black-box and data-driven models (Lepri et al., 2023).
• Real-time adaptation, hyper-reduction, and multi-scale coupling for evolving constraints and structural changes in large-scale complex networks.

The field continues to develop new algorithmic frameworks, error analyses, and computational platforms to accommodate emerging applications in control, design, and simulation of large-scale structured dynamical systems.