Model Order Reduction

• Model Order Reduction is a set of mathematical methods that condense complex dynamical systems into lower-dimensional models while preserving key system behaviors.
• It employs both projection-based techniques like POD, balanced truncation, and Krylov subspaces as well as data-driven approaches using neural surrogates for nonlinear or high-dimensional problems.
• These methods ensure computational efficiency and stability through rigorous error bounds, adaptive enrichment, and structure-preserving projections applied in simulation, optimization, control, and digital twins.

Model order reduction (MOR) refers to a collection of mathematical methods designed to approximate large-scale dynamical systems by lower-dimensional surrogate models while preserving accuracy in system behavior over quantities of interest. MOR is central to applications in simulation, design optimization, control, uncertainty quantification, and real-time digital twins, where the cost of solving the full system is prohibitive. Both projection-based and data-driven techniques have been developed, the former grounded in system theory, geometry, and numerical linear algebra, the latter integrating machine learning or data-mining to address high-dimensional, nonlinear, or parametric settings.

1. Projection-Based Model Order Reduction

The classical pipeline for MOR in linear and weakly nonlinear systems proceeds through the construction of a low-dimensional subspace that captures the essential dynamics or input-output relationships. For a state-space model

$$\dot{y}(t) = f(y(t); \mu), \quad y\in\mathbb{R}^N, \;\mu\in\mathbb{R}^{n_\mu},$$

the full trajectory is projected onto a reduced basis: $$y(t) \approx \Phi\,y_r(t),\quad \Phi \in \mathbb{R}^{N \times N_r},\; N_r \ll N.$$

The basis $\Phi$ can be constructed using methods such as:

• Proper Orthogonal Decomposition (POD): Identifies the optimal orthonormal modes (in the $\ell^2$ sense) from a "snapshot" matrix $Y$ of simulated states. The basis vectors $\{v_i\}$ are the leading left singular vectors of $Y$, where $Y=V\Sigma W^T$ and energy is measured by the singular value decay. Truncation at $N_r$ modes is typically chosen to capture a prescribed fraction of system variance (Zhuang et al., 2021).
• Balanced Truncation: For linear time-invariant (LTI) systems, solves for controllability and observability Gramians and projects onto balanced coordinates, discarding the least controllable/observable modes (Benner et al., 2020).
• Krylov Subspace and Rational Interpolation: For input-output preservation, builds rational Krylov subspaces that interpolate system frequencies; pseudo-optimal variants enforce first-order necessary conditions for the time- or frequency-limited $\mathcal{H}_2$ norm (Zulfiqar et al., 2019, Zulfiqar et al., 2022).
• Modal/Invariant Manifold Methods: For nonlinear mechanical systems, direct normal form reduction computes low-dimensional invariant manifolds in physical space, explicitly removing non-resonant couplings and capturing critical nonlinear resonances (Opreni et al., 2021).

2. Data-Driven and Non-Intrusive Model Order Reduction

Recent advances extend classical approaches using neural surrogates or data-based techniques when explicit models are unknown or highly nonlinear/system-parameter-dependent:

• Snapshot-Based Neural Surrogates: After projection by POD, the reduced system dynamics $\dot y_r$ or the discrete update $y_r(t_{j+1})$ are learned from simulated/experimental data using neural networks (Zhuang et al., 2021).
  • Direct MLP: Learns the discrete reduced map directly.
  • Runge–Kutta Neural Network (RKNN): Trains a neural network to approximate the reduced right-hand side $f_r$, then embeds it in a classic Runge–Kutta integrator, ensuring physical structure and robustness to step size.
• Randomized and Compressed Techniques: Randomized algorithms for SVD accelerate the computation of projection bases, reducing offline costs for large snapshot ensembles (Alla et al., 2016).
• Empirical Gramian and Data-Driven Balanced Approaches: For nonlinear and parametric systems, empirical simulation data is used to assemble reachability and observability Gramians or dominant subspaces (Himpe et al., 2020).

3. Time, Frequency, and Parameter Adaptivity

MOR methods are tailored to achieve specific accuracy over prescribed time, frequency, or parameter intervals:

• Time-Limited $\mathcal{H}_2$-Optimal MOR: Reduces transient (short-term) model error by defining controllability/observability Gramians on finite time horizons $[0,T]$ and constructing reduced systems to match dominant modes in that window (Zulfiqar et al., 2019).
  • Pseudo-Optimality: Formulations that satisfy a subset of $\mathcal{H}_2$ first-order conditions, delivering near-optimal transient accuracy with computational efficiency.
  • Block Greedy and TLCURE Accumulation: Adaptive, monotonic error decay by augmenting the reduced space in blocks, guaranteeing non-increasing error irrespective of interpolation choices.
• Parameter Adaptivity: For parametric or stochastic systems, reduced bases are built from sampled parameter trajectories (static/dynamic parameter sampling), or via greedy/POD in parameter space, ensuring ROM validity across a prescribed parameter domain (Mlinarić et al., 2020, Zhuang et al., 2021, Himpe et al., 2020).

4. Error Analysis, Stability, and A Posteriori Certification

MOR performance is characterized by mathematically rigorous error bounds and system-theoretic guarantees:

• Error Bounds:
  • POD and Singular Value Decay: The best-approximation error is dictated by the first neglected singular value $\sigma_{N_r+1}$, with the energy captured scaling as $\sum_{i=1}^{N_r}\sigma_i^2 / \sum_{i}\sigma_i^2$ (Zhuang et al., 2021, Alla et al., 2016).
  • $\mathcal{H}_\infty$ and $\mathcal{H}_2$ Norms: Balanced truncation and pseudo-optimal algorithms yield analytic error bounds on the transfer-function response in the respective system norms (Zulfiqar et al., 2019, Zulfiqar et al., 2022, Benner et al., 2020).
  • A Posteriori Estimators: For parametric and stochastic models, efficient residual-based error estimators ensure that online predictions remain certified within user-defined tolerances (Bontinck et al., 2017, Mlinarić et al., 2020).
• Stability and Dissipativity: Structure-preserving projections, as well as DAE-specific MOR, guarantee stability and physical consistency of the reduced models if underlying symmetry/dissipativity conditions hold (Castagnotto et al., 2015).
• Preservation of Network/Physical Structure: Special MOR techniques such as graph Laplacian (Kron) reduction for chemical networks or port-Hamiltonian projection for energy systems ensure that mass/energy conservation and positivity of states are maintained (Rao et al., 2012, Himpe et al., 2020).

5. Implementation and Computational Aspects

Practical MOR relies on scalable algorithms and software toolkits:

• Computational Kernels:
  • SVD-Based POD: Dominates the offline phase, but randomized or compressed SVD sharply reduces computational effort for very large systems (Alla et al., 2016).
  • Krylov-Based Methods: Construction of rational or block Krylov bases via shifted linear solves; small Lyapunov/Sylvester equations for coefficient extraction (Zulfiqar et al., 2019).
• Toolchains: MORLAB offers a MATLAB/Octave toolbox with modular routines for balancing, moment matching, spectral splitting, second-order and descriptor-system reduction, and adaptive error goals (Benner et al., 2020). pyMOR provides a reduction framework for parametric problems with abstract interfaces to external PDE solvers, supporting balanced truncation, IRKA, and reduced basis algorithms, with offline/online decomposition critical for fast parameter queries (Mlinarić et al., 2020).
• Greedy and Adaptive Strategies: Snapshot selection via adaptive enrichment or goal-oriented selection maximizes error decay for a given offline computational investment, particularly in high-dimensional parameter regimes or in large stochastic systems (Bontinck et al., 2017, Himpe et al., 2020).

6. Extensions and Applications

MOR techniques are adapted to diverse domains and system classes:

• Chemical Reaction Networks: Kron (Schur complement) Laplacian reduction eliminates fast complexes, preserving the graph structure and kinetic law while rigorously justifying the approximation under timescale separation (Rao et al., 2012).
• Nonlinear and Stochastic Systems: Projection and hyper-reduction (e.g., empirical interpolation or DEIM) address weakly or strongly nonlinear ODE/DAE models, including those arising from stochastic Galerkin systems via polynomial chaos (Pulch, 2017).
• Quantum and Seismic Applications: Recent advances apply projection-based MOR to quantum molecular dynamics within Kohn–Sham DFT (Cheung et al., 9 Sep 2025), open quantum systems by coarse-grained master equation expansion (Fan et al., 30 Oct 2024), and the reduction of parametric seismic wave problems by Laplace-domain projection and greedy sampling, achieving exponential convergence where snapshots exhibit band-limited behavior (Hawkins et al., 11 Jun 2024).
• Machine Learning and Deep Nets: MOR is now used both for accelerating physics-informed Neural ODEs (e.g., via POD-DEIM projection layers) (Lehtimäki et al., 2021) and for controlling the complexity of deep structured state-space models in control-centric applications, with sparsity-promoting regularization on modal/Hankel coordinates embedded within the learning objective (Forgione et al., 21 Mar 2024).

7. Current Challenges and Future Directions

Key directions and open issues in MOR research include:

• Handling Strong Nonlinearity or Nonlinear Manifolds: Classical linear subspace projection (POD) is limited for strongly nonlinear dynamics; current work explores autoencoders, nonlinear manifolds, and invariant manifold learning (Zhuang et al., 2021).
• Guarantees in Data-Driven Surrogates: Physical invariants, symmetries, and stability properties must be integrated into machine-learned ROM architectures and training (e.g., physics-informed NNs, embedded integrators, conservation constraints) (Zhuang et al., 2021).
• Time/Frequency Adaptivity and Non-Intrusive Certificates: Algorithms that efficiently target restricted time- or frequency-windows and provide certified error bounds are a focus, as are non-intrusive techniques that operate without full-system access (digital twins).
• ROMs for Large-Scale PDEs and Parametric Systems: Efficient basis compression, parameter interpolation schemes, and multi-fidelity approaches are under active development for industrial and real-time applications (Mlinarić et al., 2020, Himpe et al., 2020).
• Quantum-Scale and Open-System MOR: Direct application of MOR ideas to quantum systems---especially those with large Hilbert spaces or open system effects---is emerging, and requires new coarse-graining and regularization frameworks (Fan et al., 30 Oct 2024, Cheung et al., 9 Sep 2025).

Model order reduction remains a rapidly evolving field, combining theory from numerical linear algebra, system/dynamical systems, and data science, with a strong drive toward scalable and certified algorithms for complex and high-dimensional models across scientific and engineering domains.