SSM-Reduced Models in Dynamical Systems

• SSM-reduced models are a rigorous mathematical framework that condenses complex dynamical systems onto low-dimensional invariant spectral submanifolds.
• They employ systematic spectral analysis, Taylor expansion, and data-driven algorithms to parameterize the manifolds and derive reduced-order models.
• Explicit H2-norm error bounds and curvature-based mode selection ensure accurate, robust reductions across diverse applications such as structural mechanics and neural dynamics.

A spectral submanifold (SSM)-reduced model refers to a mathematical framework and associated algorithms for reducing high- or infinite-dimensional dynamical systems to lower-dimensional, rigorous nonlinear models supported on invariant manifolds—SSMs—associated with selected spectral subspaces of the linearized system. SSM reduction generalizes earlier invariant manifold and modal reduction techniques, providing mathematically precise reduced-order models for both linear and nonlinear, finite- and infinite-dimensional, deterministic and stochastic systems. This paradigm is applicable to domains ranging from distributed parameter control systems and structural vibrations to recurrent neural networks, finite-element models, and nonlinear time-delay systems.

1. Mathematical Foundations and Definitions

The central object in SSM-reduced modeling is the spectral submanifold: the smoothest nonlinear invariant manifold tangent at a hyperbolic fixed point (or periodic orbit) to a spectral subspace $E$ of the linearized operator $A$. In concrete terms, given a system

$\dot{x} = A x + f(x)$

with $f(x) = O(\|x\|^2)$, the SSM $W_E(0)$ is the unique $C^r$ invariant manifold tangent to $E$ at $x=0$, provided the nonresonance and spectral gap conditions are satisfied (Haller et al., 2023).

For infinite-dimensional semistable systems, where $A$ may possess a nontrivial kernel (equilibrium structure), reduction requires specialized tools such as the semistability Gramian: $$P_\infty = \int_0^\infty (S(t) - S_\infty) B B^* (S(t) - S_\infty)^* dt,$$ which satisfies the Lyapunov equation

$$A P_\infty x + P_\infty A^* x = - (I - S_\infty) B B^* (I - S_\infty)^* x.$$

This generalizes the ordinary controllability Gramian to semistable infinite-dimensional systems (Ziemann et al., 2019).

When considering nonconservative or forced finite-dimensional systems, SSMs involve time-dependent (nonautonomous) generalizations and the computation of higher-order Taylor–Fourier expansions to capture parametric resonance phenomena (Thurnher et al., 2023).

2. Construction and Computation of SSM-Reduced Models

The construction of SSM-reduced models follows a systematic workflow:

1. Spectral Analysis: Compute the spectrum of the generator (finite/infinite-dimensional matrix/operator) and select the master (slow, physically relevant, or resonant) modes.
2. SSM Parameterization: Formulate the SSM as a power series (or Taylor expansion) graph over the master coordinates, e.g.,

$$y = h(\eta) = \sum_{d=2}^M \sum_{j=0}^d a_{d,j} \eta^{d-j}.$$

The coefficients are computed recursively by matching the invariance equation at each order.

1. Reduced Dynamics: Extract the reduced-order polynomial ODE (or PDE/DAE) for the master coordinates using invariance,

$\dot{\eta} = W_r \phi(\eta),$

where $\phi(\eta)$ collects all polynomial terms up to a prescribed order and $W_r$ is the coefficient matrix.

1. Handling Constraints and Non-Smoothness: For systems with algebraic constraints (e.g., in multibody mechanics) or non-smooth switching elements, the SSM construction is performed in each smooth subregion and then “patched” using projection formulas to transition across the switching surface (Bettini et al., 2023, Li et al., 2022).

For infinite-dimensional systems, model reduction is enabled under a commutativity assumption: projection and embedding operators $(\pi, \sigma)$ are constructed so that $\hat{A} \pi x = \pi A x$ ensures semigroup-level commutativity and preservation of semistability, yielding exact a priori $\mathcal{H}_2$-norm error formulas (Ziemann et al., 2019).

In data-driven contexts, algorithms such as SSMLearn, fastSSM, and fastSSM$^+$ fit the SSM and its reduced dynamics from observed time-series by singular value decomposition, polynomial regression, and sparse normal-form transformation, significantly accelerating computations versus global implicit optimization (Axås et al., 2022).

3. Error Quantification and Mode Selection

A central feature of SSM-based reduction is the capacity for rigorous error estimation and systematic basis selection:

• $\mathcal{H}_2$-Norm Error Bounds: For infinite-dimensional semistable systems with invariant reductions,

$$\| \Sigma - \hat{\Sigma} \|_{\mathcal{H}_2} = \sqrt{ \operatorname{tr} (C (I - \sigma \pi) P_\infty (I - \sigma \pi)^* C^*) },$$

providing explicit, computable measures of how well the reduced model captures the system’s energy response (Ziemann et al., 2019).

• Curvature-Based Mode Selection: For large nonlinear mechanical systems, mode sets are augmented by including slave modes whose associated SSMs exhibit locally large curvature. The directional scalar curvature $curv_k(I)$ is evaluated via explicit tensorial formulas involving the quadratic SSM coefficients, with only those modes whose nonlinear influence exceeds a threshold being retained (Buza et al., 2020).

These capabilities move reduction methods beyond ad hoc or heuristic selection, providing algorithmic foundations for ensuring accuracy and relevance.

4. Extensions: Fractional and Mixed-Mode SSMs, Random and Time-Delay Systems

Classical SSM reduction assumed master spectral subspaces of like stability type and integer-powered expansions. Recent theoretical advances rigorously establish the existence and construction of:

• Fractional and Mixed-Mode SSMs: For arbitrary spectral subspaces (including those containing unstable and stable modes), an extended family of (not necessarily analytic) invariant manifolds exists. Their expansions generically involve fractional powers of the master variables, crucial for capturing transitions (heteroclinic orbits, buckling, turbulence onset) and emergent multi-attractor phenomena (Haller et al., 2023).
• Random SSMs: In systems with bounded random forcing, the deterministic SSM persists as a “random invariant manifold” solving a cocycle-based stochastic problem, enabling Monte Carlo computation of statistical quantities such as power spectral densities in large nonlinear mechanical systems at dramatically reduced computational cost (Xu et al., 4 Jul 2024).
• Time-Delay System Reduction: Nonlinear time-delay systems are approximated as high-dimensional ODEs via chain discretization. SSMs are then computed for the ODE system, yielding ROMs that capture bifurcations, isolas, and quasi-periodic orbits in the original delay system (Tang et al., 6 May 2025).

5. Practical Implementation, Non-Intrusive and Data-Driven Approaches

SSM-reduced models are implemented in both equation-driven and data-driven settings:

• Non-Intrusive Algorithms: For finite-element models lacking explicit access to nonlinear coefficients (as with commercial solvers), non-intrusive algorithms couple black-box nonlinearity evaluation (via the STEP method) with the parameterization approach, decomposing nonlinear forces into even/odd parts and leveraging only real-valued evaluations. This allows SSM reduction in highly complex mechanical systems (MEMS, plates, shells) (Li et al., 16 Sep 2024).
• Data-Driven Construction: SSMLearn and its variants construct SSMs directly from time-series data, using SVD to identify the slow tangent space, polynomial regression for manifold fitting, and recursive normal-form computation. Even with sparse, unforced transient data, these methods yield ROMs that can predict forced response and bifurcations with high accuracy, crucial for applications where full simulation is infeasible (Cenedese et al., 2023).
• Oblique Projection for Non-Normal Systems: In systems with strong non-normality, an oblique projection along the stable invariant foliation (rather than a normal projection) better aligns off-manifold trajectories to their correct reduction coordinates, enhancing accuracy even with minimal data (Bettini et al., 27 Mar 2025).
• Software and Computational Tools: Open-source packages such as SSMTool automate high-order, non-autonomous SSM computation, backbone and forced response extraction, error estimation, and continuation-based bifurcation analysis for both ODEs and constrained multibody systems (Li et al., 2022, Thurnher et al., 2023).

6. Applications in Engineering, Control, and Data Science

Representative applications across domains include:

• Distributed Parameter and PDE Systems: Heat equations with boundary conditions leading to nontrivial kernels, controlled systems with spatially distributed states, and scenarios with approximate controllability and stability structure (Ziemann et al., 2019).
• Structural and Multibody Mechanics: Reduced modeling of beams, plates, shells, and constrained multibody systems, capturing backbone and forced response curves, internal resonances, and configuration constraints (Veraszto et al., 2019, Li et al., 2022, Li et al., 16 Sep 2024).
• Neural Dynamics and RNNs: Extraction of mathematically precise, low-dimensional (line, ring attractor) models explaining decision boundaries and working-memory patterns in artificial RNNs and neuronal data (Marraffa et al., 15 Oct 2025).
• Model Reduction in SSM-Based LLMs: For deep SSM-based architectures (e.g., Mamba, Zamba), reduction and pruning strategies (OBS, H$^2$ optimization, token reduction) target state-space parameters for deployment efficiency, inference speed, and adaptive resource usage (Tuo et al., 11 Jun 2025, Sakamoto et al., 14 Jul 2025, Zhan et al., 16 Oct 2024, Forgione et al., 21 Mar 2024, Glorioso et al., 26 May 2024).

These reductions enable applications ranging from model-based control, uncertainty quantification, and optimal design to rapid forced-response simulation and advanced dynamics identification—even for structurally complex, high-dimensional, or partially observed systems.

7. Theoretical and Computational Advantages, Limitations, and Outlook

SSM-reduced models exhibit several rigorous and practical strengths:

• Uniqueness and Maximal Smoothness: The primary SSM is unique and the smoothest invariant manifold tangent to a given spectral subspace, in contrast to generally non-unique center manifolds. For forced and stochastic systems, this smoothness and persistence are maintained under mild perturbations (Haller et al., 2023, Xu et al., 4 Jul 2024).
• Error Analysis and Robustness: Explicit $\mathcal{H}_2$ error bounds, curvature-based mode inclusion criteria, and built-in invariance error diagnostics support robust model selection and validation (Ziemann et al., 2019, Buza et al., 2020).
• Flexibility Across Modalities: Both intrusive (coefficient-level) and non-intrusive (black-box, data-driven) versions exist. Algorithms can reach arbitrary Taylor order and efficiently handle high-dimensional or constrained domains.
• Scalability: Demonstrated speed-ups of several orders of magnitude compared to full simulations for both autonomous and externally forced problems, including those involving millions of degrees of freedom in finite-element settings (Cenedese et al., 2023, Li et al., 16 Sep 2024).
• Generalization to Non-Smooth, Random, and Delay Systems: By leveraging patched manifold constructions and random invariant manifold theory, SSM reduction extends beyond the classical analytic/smooth deterministic ODE setting.

Limitations include computational cost for very high-order expansions, sensitivity to near-resonances (affecting inversion of near-singular operators), and the need for careful tuning/validation of data-driven fitting in noisy or sparse-data environments. For non-normal and high-dimensional systems, oblique or nonlinear projections may be necessary to avoid systematic reduction artifacts.

Further avenues include structured model order reduction in deep SSM architectures, hardware-friendly pruning for resource-constrained inference, and extensions to Bayesian and stochastic contexts.

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In summary, SSM-reduced modeling constitutes a rigorously founded and computationally versatile set of methodologies for the principled reduction of complex dynamical systems—enabling the extraction, analysis, and simulation of their essential nonlinear (and stochastic, delayed, or constrained) behaviors on lower-dimensional invariant manifolds across scientific and engineering domains.