Dynamic Mode Decomposition in Nonlinear Dynamical Systems

Dynamic Mode Decomposition (DMD) has emerged as a powerful, data-driven technique for extracting coherent spatio-temporal structures from complex, high-dimensional systems. Originally conceived for near-linear dynamics, DMD approximates the action of the infinite-dimensional Koopman operator by analysing time-ordered snapshots of system observables. By performing a singular value decomposition on snapshot matrices and fitting a low-rank linear operator, DMD yields a set of eigenvalues and modes that characterise growth rates, oscillation frequencies and spatial structures inherent to the underlying dynamics. In nonlinear settings, variants such as extended DMD, Hankel-DMD and kernel-based DMD embed nonlinear observables or delay coordinates into the analysis to capture richer dynamics. These adaptations allow practitioners to construct reduced-order models that faithfully reproduce key features of chaotic fluid flows, neural activity, epidemiological outbreaks and climate phenomena. The ability to isolate dominant modes, forecast future states and inform control strategies has made DMD a cornerstone of modern dynamical systems research, bridging the gap between machine learning, operator theory and practical applications.

Research from Nature Portfolio

Recent studies have advanced the framework of data-driven modal decompositions. A novel combination of a β-variational autoencoder and a transformer architecture has been introduced to learn compact, near-orthogonal latent representations of complex fluid flows. This approach identifies features analogous to classical modal bases while outperforming existing predictors in both periodic and chaotic regimes, demonstrating potential for weather forecasting and structural dynamics. Foundational work on deep learning for Koopman embeddings has leveraged modified autoencoders to discover intrinsic coordinates in which strongly nonlinear systems evolve linearly, enabling global linearisation and interpretable spectral decompositions. Earlier contributions to Hankel alternative view of Koopman (HAVOK) analysis have shown that chaotic dynamics can be decomposed into a linear system forced intermittently by low-energy modes, thereby revealing coherent linear regimes and pinpointing rare forcing events that precede abrupt transitions.

Research from all publishers

Extensions of DMD beyond major portfolios have led to robust control-aware and kernel-based methodologies. Dynamic Mode Decomposition with Control (DMDc) disambiguates intrinsic dynamics from external actuation by incorporating input–output data, yielding accurate low-order models for systems under control without prior knowledge of governing equations. Kernel-based Koopman spectral analysis employs implicit feature maps defined by user-specified kernels to span richly nonlinear observable spaces, improving approximation of spectral quantities in high-dimensional problems while avoiding prohibitive computational costs. Complementary approaches have blended classical numerical integration schemes with reservoir computing: Runge–Kutta guided next-generation reservoir computing combines physics-informed time stepping with data-driven recurrent networks to predict and identify governing equations of chaotic systems, offering an interpretable alternative to traditional modal decompositions.

Dynamic Mode Decomposition in Nonlinear Dynamical Systems publication trend

The graph below shows the total number of articles in dynamic mode decomposition in nonlinear dynamical systems across all publications each year (not limited to Nature Index journals).

Technical terms

Dynamic Mode Decomposition (DMD): A data-driven algorithm that approximates the spectral properties of the Koopman operator by analysing time-series snapshots to extract eigenvalues and spatial modes.

Koopman operator: An infinite-dimensional linear operator that evolves observable functions of a dynamical system, providing a framework for global linearisation of nonlinear dynamics.

Reduced-order model: A low-dimensional representation of a high-dimensional system obtained by projecting its behaviour onto a small number of dominant modes or coordinates.

Observable function: A scalar or vector function of the system state used to lift nonlinear dynamics into a space amenable to linear analysis.

Hankel embedding: A technique that constructs delay-coordinate matrices from time-series data, enabling the capture of temporal correlations and hidden dynamics in DMD analyses.

References

1. β-Variational autoencoders and transformers for reduced-order modelling of fluid flows. Nature Communications (2024).
2. Deep learning for universal linear embeddings of nonlinear dynamics. Nature Communications (2018).
3. Chaos as an intermittently forced linear system. Nature Communications (2017).
4. Dynamic Mode Decomposition with Control. SIAM Journal on Applied Dynamical Systems (2016).
5. A kernel-based method for data-driven koopman spectral analysis. Journal of Computational Dynamics (2015).