Block Structure Preserving MOR

• Block structure preserving model order reduction is a methodology that compresses large-scale dynamical systems by maintaining inherent block structures to enforce physical laws.
• Projection-based, clustering, and optimization-driven techniques enable reduced-order models that uphold conservation laws and provide rigorous error bounds.
• These approaches enhance computational efficiency and stability in multi-physics and networked systems by explicitly preserving inter-field couplings.

Block structure preserving model order reduction (MOR) comprises a family of algorithms and methodologies designed to reduce large-scale dynamical systems by compressing their degrees of freedom while explicitly retaining the partitioned or coupled structure inherent to the original problem. These approaches target systems whose governing equations or matrices exhibit block form due to multiple physical fields, algebraic constraints, inherent network architectures, or coupled multi-physics. By preserving partitions such as dynamical/algebraic, current/potential, or inter-physics couplings, block structure preserving MOR enables the reduced-order models (ROMs) to maintain physical interpretability, enforce conservation laws or constraints, and yield superior accuracy and computational efficiency compared to monolithic reductions. The approaches span projection-based reduction, interpolation and clustering, optimization-driven schemes, and data-driven operator inference.

1. Conceptual Basis and Motivation

Block structure arises naturally in many scientific and engineering models, including:

• Multi-field PDEs and integral equations: Coupled electric/magnetic, fluid/structure, or port-Hamiltonian formulations
• DAEs and descriptor systems: Partitioning into dynamical and algebraic components (e.g., semi-explicit index-1 DAEs)
• Physical networks: Graph-based interconnections, e.g., mass–spring–damper systems, biochemical reaction networks, and hydraulic or electrical grids
• Operator inference: Segregating operators and nonlinearities between physics components in coupled models

Preserving block structure ensures that the ROM respects physical laws (e.g., charge conservation, energy balance), honors inter-field couplings, and supports further coupling in modular simulation frameworks. For instance, in the Augmented Electric Field Integral Equation (A-EFIE) system, the unknown electric current and charge distributions are coupled by continuity and inductive/resistive physics, leading to a block system; reduction that respects this structure supports accurate recovery of both fields and constraint satisfaction (Torchio et al., 17 Nov 2025).

2. Projection-Based Block-Preserving Reduction

Projection-based MOR methods construct separate or coupled trial/test subspaces that reflect the block organization of the original system. Examples include:

• A-EFIE Integral Equations: The full system after Galerkin discretization reads

$$A x = \begin{pmatrix} R + i\omega L & S^T \ P S & -i\omega I \end{pmatrix} \begin{pmatrix} j \ \phi \end{pmatrix} = b,$$

where $j$ (currents) and $\phi$ (scalar potentials/charges) reside in distinct subspaces. Block-structure preserving MOR forms two reduced bases $V_1, V_2$ for $j$ and $\phi$ using frequency-sweep snapshots and Gram–Schmidt orthogonalization. The reduced matrices are assembled blockwise:

$$\begin{aligned} \tilde{A}_{11}(\omega) &= V_1^T [R + i\omega L] V_1, \ \tilde{A}_{12} &= V_1^T S^T V_2,\qquad \tilde{A}_{21} = V_2^T P S V_1, \ \tilde{A}_{22}(\omega) &= -i\omega (V_2^T V_2). \end{aligned}$$

This approach ensures retained couplings and better error/constraint control than a monolithic projection (Torchio et al., 17 Nov 2025).

• Semi-Explicit Index–1 DAE Reduction: For systems

$$\begin{bmatrix} E_{11} & 0 \ 0 & 0 \end{bmatrix} \begin{bmatrix} \dot{x}_1 \ \dot{x}_2 \end{bmatrix} = \begin{bmatrix} A_{11} & A_{12} \ A_{21} & A_{22} \end{bmatrix} \begin{bmatrix} x_1 \ x_2 \end{bmatrix} + B u,$$

block-structure preserving projection reduces only the dynamic subspace $x_1$, keeping $x_2$ (the algebraic variables) intact, via one-sided Krylov subspaces or balanced truncation. The ROM remains a block-partitioned system, thus maintaining algebraic constraints and asymptotic stability (Castagnotto et al., 2015).

• Port-Hamiltonian and Second Order Systems: Petrov–Galerkin projections with structure-adapted test spaces (e.g., $W = Q V$ for port-Hamiltonian with energy matrix $Q$) preserve skew-symmetry and definiteness of the system matrices, guaranteeing the reduced system remains port-Hamiltonian, passive, and stable (Mamunuzzaman et al., 2022).

3. Block-Preserving Reduction via Clustering and Network Partitioning

Clustering-based MOR constructs ROMs by regrouping states according to network or graph structure, exploiting the fact that many complex networks admit coarse representations whose nodes embody entire clusters of the original vertices.

• Physical Network Systems: Given a graph $\mathcal{G}=(V,E)$ with weighted Laplacian $L$ and vertex weights $M$, an almost equitable partition (AEP) $P$ defines clusters with preserved interconnection weights. The reduced system is constructed as

$$\dot{\hat{x}} = -\hat{D} \hat{R} \hat{D}^T \hat{M}^{-1} \hat{x} + \hat{E} u,$$

where $\hat{D}, \hat{R}, \hat{M}$, and $\hat{E}$ are coarse-grained analogues of $D, R, M, E$ obtained via $P^T$-projections. With AEP, the transformation renders the system matrix block-diagonal, and the clustered system accurately represents inter-cluster interactions while providing a computable $H_2$–norm error bound (Monshizadeh et al., 2014).

• Biochemical Networks: Block-diagonal Gramians are found that respect the chosen network or species partition. The balancing/projection yields a reduced model that is stable, positive, and maintains the original network’s interconnection pattern, with explicit error bounds and two reduction schemes: full balanced truncation and steady-state truncation (Sootla et al., 2014).

4. Optimization-Based and Data-Driven Block Structure Preservation

Recent advances employ optimization or data-driven techniques to impose block structure directly in reduced models.

• Structured Optimization-Based MOR (SOBMOR): The reduced model matrices (e.g., for pH or symmetric second-order forms) are parameterized to satisfy block-structural and algebraic constraints by construction. The optimization target is a (smooth) frequency response mismatch—usually a leveled least-squares functional over sampled frequencies. The parameter space includes only structure-admissible candidates; block- or sparsity-patterns may be explicitly encoded. Quasi-Newton (BFGS) algorithms are used to minimize the objective, and the approach yields ROMs with orders-of-magnitude smaller $H_\infty$ errors than classical projection-based block-preserving methods, without compromising stability or passivity (Schwerdtner et al., 2020).
• Block-Structured Operator Inference: Operator Inference is extended to multi-physics systems by partitioning the reduced-order state and the induced inference problems according to each physics component and their coupling terms. The methodology proceeds by collecting separate projections (e.g., via POD) for each block, and learning block-coupled ROM operators via regularized least-squares. This block structure ensures property preservation (stability, energy balance, second-order structure), enables tailored regularization, and results in ROMs with reduced parameter complexity. In aeroelastic benchmarks, block-structured OpInf achieves 20% online prediction speedup with parity or superior accuracy compared to monolithic operator inference (Zastrow et al., 7 Nov 2025).

5. Structure-Preserving MOR for PDEs, Descriptor Systems, and Integral Equations

Block structure preservation plays a crucial role in more complex settings:

• Boundary-Controlled 1D PH Systems: Partitioned finite element discretization constructs FOMs in block port-Hamiltonian form, and structure-preserving Loewner framework reduction via Petrov–Galerkin biorthogonal projections yields ROMs that retain passivity, block interconnections, and physical energy interpretation. Such methodologies are applicable to hyperbolic PDEs with boundary ports—waves and beams with general boundary conditions (Toledo-Zucco et al., 9 Feb 2024).
• Integral Equation Formulations (A-EFIE): Block-structure preserving MOR reduces the number of required modes and snapshots while substantially lowering physical constraint residual (e.g., divergence/solenoidality errors) compared to monolithic reduction. The methodology extends to other block-coupled or multi-physics integral schemes (Torchio et al., 17 Nov 2025).

6. Performance, Error Bounds, and Applicability

Block-structure preserving MOR yields ROMs that, compared to monolithic or unstructured reduction, exhibit:

• Lower order and enhanced accuracy for a given output tolerance (e.g., fewer snapshots/basis vectors in electromagnetic integral equation reduction (Torchio et al., 17 Nov 2025))
• Strict preservation of physical constraints (e.g., conservation laws, continuity, passivity, stability), as the blockwise or partitioned nature of projections or optimization aligns with the invariants encoded in the original block couplings
• Explicit error measures and analytical bounds (e.g., $H_2$ bounds for clustering-based reductions (Monshizadeh et al., 2014), gap metrics for structured balanced truncation (Dorschky et al., 2020, Breiten et al., 2021))
• Computational advantages via reduced basis dimensions, specialized regularization, and modularity

Some approaches (e.g., SOBMOR (Schwerdtner et al., 2020)) for port-Hamiltonian and symmetric second-order systems yield ROMs that outperform projection-based block-MOR schemes by 2–3 orders of magnitude in $H_\infty$ and $H_2$ error.

The methodology applies broadly: to port-Hamiltonian, descriptor, symmetric second-order systems, DAEs, PDEs with energy or constraint structure, networked multi-agent and biochemical models, electromagnetic integral formulations, and multiphysics operator inference.

7. Limitations and Practical Considerations

Block-structure preserving MOR introduces additional algorithmic overhead due to separate basis construction, bookkeeping, and parameterization. For high-dimensional or highly coupled systems, greedy sampling or operator inference may still be expensive. Nevertheless, the significant reductions in reduced-model order, error, and physical residuals justify the complexity in most applications. Extensions to hybrid, multi-domain, and nonlinear settings are natural—especially wherever the full-order operator decomposes into coupled, physically distinct blocks.

A plausible implication is that ongoing research will focus on automating the partition/basis selection process, exploiting data/simulation pipelines and integrating optimization-based block constraints with machine-learned models, to further extend block-structure preserving MOR to real-time, nonlinear, and black-box systems.