Damped Linear Oscillatory State-Space Models

• D-LinOSS are state-space models that integrate explicit damping and oscillatory behavior to capture both transient decay and periodic dynamics.
• They employ independent spectral parameterization to flexibly position eigenvalues, enhancing control over memory timescales and stability.
• The models support efficient, stable learning applicable to diverse fields such as time series forecasting, sensor regression, and control systems.

A Damped Linear Oscillatory State-Space Model (D-LinOSS) is a state-space model whose internal dynamics are driven by linear, damped, and potentially oscillatory evolution. This class of models generalizes conventional linear state-space models by incorporating explicit damping (energy dissipation) alongside oscillatory (complex-conjugate eigenvalue) behavior and enables both theoretical analysis and efficient, stable implementation in applications ranging from time series forecasting to large-scale sequence modeling.

1. Mathematical Structure and Model Definition

A prototypical D-LinOSS is based on the dynamics of the damped harmonic oscillator, described by the second-order ordinary differential equation (ODE): $\ddot{x}(t) + g\, \dot{x}(t) + a\, x(t) = b\, u(t)$ where $g \ge 0$ is the damping coefficient, $a \ge 0$ controls the oscillation frequency, $u(t)$ is the input or forcing, and $b$ is the input gain. For vector-valued state $x \in \mathbb{R}^m$, $g$ and $a$ are typically diagonal or block-diagonal matrices.

This ODE is recast into first-order state-space form by introducing $z(t) \equiv \dot{x}(t)$: $$\begin{bmatrix} \dot{x}(t) \ \dot{z}(t) \end{bmatrix} = \begin{bmatrix} 0 & I \ -a & -g \end{bmatrix} \begin{bmatrix} x(t) \ z(t) \end{bmatrix} + \begin{bmatrix} 0 \ b \end{bmatrix} u(t)$$ For practical deep learning and signal processing implementations, a discrete-time discretization is used: $$\begin{aligned} z_{k+1} &= z_k + \Delta t \left[ -a\, x_k - g\, z_{k+1} + b\, u_{k+1} \right] \ x_{k+1} &= x_k + \Delta t\, z_{k+1} \end{aligned}$$ or, after algebraic manipulation and parameterization: $$w_{k+1} = M\, w_k + F\, u_{k+1}\,,\quad y_{k+1} = H\, w_{k+1}$$ where $w = [x; z]$ is the stacked state and $M$ is a $2m \times 2m$ recurrent matrix whose spectral properties are determined by the learned $a$ and $g$. The output $y_k$ is typically a linear function of $w_k$.

The eigenvalue spectrum of $M$ is fully characterized by $a, g, \Delta t$ per latent dimension, yielding complex-conjugate pairs inside (or on) the unit disk; real parts (damping), imaginary parts (oscillation frequency), and modulus (memory timescale) are learned or specified per dimension.

2. Model Expressivity, Universality, and Spectral Parameterization

D-LinOSS generalizes prior oscillatory state-space models by introducing a trainable, decoupled damping term:

• In earlier LinOSS models, dissipation (energy loss, i.e., eigenvalue modulus) is determined rigidly by the discretization (implicit or IMEX), coupling decay to oscillation via a fixed functional relationship.
• In D-LinOSS, $g$ (the damping matrix) is independently learnable and parameterized, enabling a precise control of the spectral radius for each latent state dimension and consequently allowing a broader, more flexible range of temporal decay and oscillatory behaviors (Boyer et al., 17 May 2025).

This additional flexibility yields:

• The ability to place eigenvalues almost anywhere in the unit disk (versus being confined to a circle or ring in the complex plane).
• Spectral bijection: for any target eigenvalue $\lambda = re^{i\omega}$ with $|r|\leq 1$, there is a unique $(a, g, \Delta t)$ triple mapping to it.
• Universal approximation: D-LinOSS (like undamped LinOSS) is proven to approximate any continuous causal operator between time series to arbitrary accuracy, with universality not compromised by the introduction of damping (Rusch et al., 4 Oct 2024).

Empirically, this leads to accurate modeling of sequences exhibiting exponential decay, long memory, and a continuum between periodic and over-damped trajectories.

3. Stability and Learning Dynamics

Stable dynamics in D-LinOSS require that all eigenvalues of $M$ stay inside the unit disk (i.e., $|\lambda|\le1$) in the discrete setting. Sufficient conditions are: $(g_i - \Delta t\, a_i)^2 \leq 4 a_i$ for each decoupled oscillator $i$, where $a_i$ and $g_i$ are the (possibly per-state) effective frequency and damping.

In training pipelines, $g$ and $a$ are often parameterized as $g = \text{ReLU}(\hat{g})$ and $a = \text{Clamp}(\hat{a}, L(g), U(g))$ where $L(g), U(g)$ are bounds arising from spectral constraints, ensuring stable updates and avoiding parameter drift into chaotic or unstable regimes (Boyer et al., 17 May 2025).

D-LinOSS supports efficient parallelization and $O(m)$ per-timestep complexity, as all recurrent updates are performed with diagonal or block-diagonal matrices. Initialization is made robust by inverting the eigenvalue-to-parameter mapping, allowing pre-specification of eigenvalue spectra (e.g., logarithmic ladder of decay rates and frequencies).

4. Connections to Classical and Statistical Modeling Frameworks

Relation to Classical System Theory and Control

D-LinOSS generalizes canonical state-space models used in control theory, signal processing, and physics, e.g., the damped harmonic oscillator, and directly connects to models such as:

• Dynamic Linear Models (DLMs) with damped oscillatory states, a standard construction in time-series analysis, geodesy, and engineering (Laine, 2019).
• Modal analysis in stochastic dynamical systems, where the system matrix can be block-diagonalized into real and complex modes, allowing explicit estimation and interpretation of damping rates, frequencies, and mode shapes (MacKay, 2019).
• DSGE models in economics, where under-damped, critically damped, and over-damped responses are critical for simulating delayed or cyclical adjustment after shocks (Hsu, 10 Feb 2025).
• Physical PDEs and distributed systems, where D-LinOSS serves as a finite-dimensional approximation, particularly when coupled with Lyapunov analysis for stability and control (Chitour et al., 2023).

Spectral Perspective and Gaussian Process Equivalence

The D-LinOSS prior encompasses the state-space (SDE) equivalent of stationary or stationary-oscillatory Gaussian processes, including Matérn kernels with $\nu>1/2$, and the damped simple harmonic oscillator, with Kalman filtering enabling $O(n)$ inference and learning (Jordán et al., 2021).

5. Practical Applications and Extensions

D-LinOSS models are empirically validated on tasks as diverse as:

• Long-sequence time series classification (e.g., UEA benchmarks), where D-LinOSS yields empirical gains in accuracy over LinOSS variants and attention models for sequences of up to 50k tokens (Boyer et al., 17 May 2025).
• Real-world sensor data regression (e.g., heart rate estimation, weather forecasting), requiring both long memory and adaptive forgetting.
• Synthetic tasks requiring controlled exponential decay and selective recall across varying damping and frequency regimes.
• Selection and gating mechanisms, where D-LinOSS's stable and interpretable internal states allow design of residual-based selectors analogous to fault detection systems in control (Casti et al., 23 May 2025).

Applications in physical modeling (seismic, climate, biophysical, etc.), computational neuroscience, and financial engineering benefit from the capacity to model slow or fast decay, coherent oscillatory bursts, and transient phenomena, with flexible learning of decay characteristics.

6. Bayesian and Inference Methodologies

The D-LinOSS framework is compatible with both frequentist (maximum likelihood, EM) and Bayesian approaches:

• Variational Bayesian inference and Kalman filtering can be deployed for time-varying, hierarchical (e.g., switching or smoothly varying) D-LinOSS models, as in (Luttinen et al., 2014).
• Mode decomposition and covariance-based Bayesian techniques enable high-fidelity recovery of dominant damped oscillatory modes, validation, and uncertainty quantification, particularly in settings with low SNR (Pumpe et al., 2016, MacKay, 2019).

Extensions to handle singular process noise, as required in some physical and economic models, are efficiently supported by latent disturbances and convex relaxation (via Lagrangian relaxation and semidefinite programming), safeguarding convergence and stability (Umenberger et al., 2016).

7. Advanced Topics: Nonlinear, Time-Varying, and Stochastic Extensions

D-LinOSS formulations admit several advanced generalizations:

• Nonlinear and time-varying dynamics, including systems whose “modes” or “damping” smoothly evolve via latent Gaussian processes or linear combinations of dynamic matrices (Luttinen et al., 2014).
• Explicit treatment of non-resonant, slowly decaying or stochastic perturbations using averaging and Lyapunov techniques, which yield “averaged” amplitude equations and robust stability guarantees (Sultanov, 2023, Sultanov, 2023).
• Determination and tracing of “critical damping surfaces” in parameter space via ODE-based continuation, supporting design and calibration of systems at the edge between oscillatory and aperiodic regimes (Lázaro, 2018).

Such advanced techniques permit D-LinOSS models to capture a much wider spectrum of dynamical behaviors and adapt to varying stability regimes, noise conditions, and parameter drift.

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Summary Table: Key Features of D-LinOSS

Feature	Details / Mathematical Representation	Significance
Core ODE	$\ddot{x} + g \dot{x} + a x = b u$	Explicit damping and oscillation
Discrete State Update	$w_{k+1} = M w_k + F u_{k+1}$, with flexible $M$ spectrum	Controllable decay and memory timescales
Spectrum Parameterization	Bijection: $(a, g, \Delta t) \leftrightarrow \lambda$, $\lambda$ in unit disk	Universal coverage of stable/oscillatory behaviors
Learning/Initialization	Direct spectral initialization; stability via parameter reparameterization and clamping	Robust and efficient training
Inference Methods	Kalman filter, MCMC, Variational Bayes, Latent Disturbance EM, Convex SDPs	Bayesian, scalable, robust to low SNR/singularities
Application Domains	Long-sequence modeling, time-series regression, physics, economics, fault detection mechanisms	High accuracy and interpretability in broad settings

D-LinOSS models represent the current theoretical and practical frontier in oscillatory state-space modeling, combining stable and expressive dynamics with scalable, interpretable, and highly effective architectures for both scientific and deep learning contexts.