Dynamic Mode Decomposition

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2 papers in the last four weeks, down 33% on the four weeks before. 0.0% of all new papers.

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Latest papers 27

Sep 30, 2026eess.SP

PI-AMFM: Permutation-Invariant Learning for Variable-Cardinality AM-FM Mode Decomposition in Biomedical Signal Analysis

Physiological recordings often contain nonstationary oscillatory components whose number and dynamics vary across signals. Amplitude- and frequency-modulated (AM-FM) representations are well suited to characterizing such dynamics and have shown broad utility in biomedical signal analysis. Recent approaches have incorporated neural networks to learn mode decomposition patterns from data, but component cardinality is often predefined or determined through separate stopping or selection mechanisms. We propose a permutation-invariant neural framework for variable-cardinality AM-FM mode decomposition (PI-AMFM). PI-AMFM combines a multiscale temporal encoder, Mamba backbone, and component-presence estimation, with permutation-invariant Hungarian matching during training. On synthetic AM-FM signals, PI-AMFM achieved lower decomposition, instantaneous-frequency, reconstruction, and mode-count errors than the compared methods while preserving the overall trajectory pattern in a crossing-chirp example. On photoplethysmographic recordings, recovered modes captured cardiac and respiratory dynamics despite training only on synthetic signals. These results support the feasibility of PI-AMFM for variable-cardinality decomposition of nonstationary biomedical signals.

Youngsun Kong, Ki H. Chon

Biomedical Engineering, University of Connecticut, Storrs, CT 06029, USA

Dynamic Mode Decomposition Physiological Signals Time-Frequency Frequency Neural Recordings Equivariant Learning+1

Sep 24, 2026math.DS

Functional dynamic mode decomposition: Learning infinite-dimensional systems from data

Dynamic mode decomposition (DMD) is a data-driven method that computes the best linear approximation of the underlying dynamical system and decomposes the dynamics into a superposition of characteristic spatiotemporal patterns. Originally introduced by the fluid dynamics community, DMD and its extensions have found widespread use in many other research areas such as molecular dynamics, climate science, engineering, finance, and neuroscience. Applications include dimensionality reduction, forecasting, system identification, control, and spectral clustering. In order to apply DMD to partial differential equations, the spatial domain is typically first discretized using finite difference or finite element techniques, thus implicitly rendering the problem finite-dimensional. We extend projected and exact DMD to infinite-dimensional systems. Rather than estimating matrices from vector-valued observations, our DMD variants learn finite-rank operators from functional data such as observables, densities, or wavefunctions. We show that conventional DMD algorithms can be regarded as special cases of their functional DMD counterparts. All results will be illustrated with the aid of guiding examples. We focus in particular on Koopman, Perron-Frobenius, and Koopman-von Neumann operators associated with graphons, ordinary differential equations, and stochastic differential equations.

Stefan Klus, Eirini Ioannou

School of Mathematical & Computer Sciences, Heriot–Watt University, Edinburgh, UK · Maxwell Institute for Mathematical Sciences, University of Edinburgh and Heriot–Watt University, Edinburgh, UK

Dynamic Mode Decomposition Dynamical Systems Stochastic Differential Equations Decomposition

Sep 15, 2026cs.LG

High-Fidelity Digital Twin Data Models by Randomized Dynamic Mode Decomposition and Deep Learning with Applications in Fluid Dynamics

The purpose of this paper is the identification of high-fidelity digital twin data models from numerical code outputs by non-intrusive techniques (i.e., not requiring Galerkin projection of the governing equations onto the reduced modes basis). In this paper the author defines the concept of the digital twin data model (DTM) as a model of reduced complexity that has the main feature of mirroring the original process behavior. The significant advantage of a DTM is to reproduce the dynamics with high accuracy and reduced costs in CPU time and hardware for settings difficult to explore because of the complexity of the dynamics over time. This paper introduces a new framework for creating efficient digital twin data models by combining two state-of-the-art tools: randomized dynamic mode decomposition and deep learning artificial intelligence. It is shown that the outputs are consistent with the original source data with the advantage of reduced complexity. The DTMs are investigated in the numerical simulation of three shock wave phenomena with increasing complexity. The author performs a thorough assessment of the performance of the new digital twin data models in terms of numerical accuracy and computational efficiency.

Diana A. Bistrian

University Politehnica Timisoara, Department of Electrical Engineering and Industrial Informatics, Romania

Digital Twins Dynamic Mode Decomposition Fluid Dynamics Shock Propagation Deep Learning

Aug 13, 2026cs.AI

DMDIntel: Interpreting Large Language Models via Dynamic Mode Decomposition

In this work, we introduce DMDIntel which uses dynamic mode decomposition (DMD) to make the predictions made by LLMs in a classification task interpretable. It develops an input attribution pipeline, that first decomposes the hidden states of an LLM into prominent patterns, also known as modes, and then associates ranks to the input tokens based on the projection values on those modes. Rigorous experiments across three datasets and three model families consistently show that the ranked attribution of input tokens obtained using DMDIntel by far outperforms state-of-the-art techniques such as principal component analysis, integrated gradients and SHAP.

Amogh Joshi, Animesh Mukherjee, Sergey Utyuzhnikov

IIT Kharagpur, India · University of Manchester, UK

Decomposition Dynamic Mode Decomposition Nvidia Principal Component Analysis

Aug 5, 2026cs.LG

IMFACT: Counterfactual Explanations for Time Series via Intrinsic Mode Function Substitution

Oscillatory signals, such as vibration, carry class-discriminative information in specific frequency bands; perturbing them in raw feature space for counterfactual analysis easily destroys their temporal structure and produces physically implausible results. In this work, we introduce IMFACT (IMF-based counterfACTuals), a model-agnostic framework for generating plausible counterfactual explanations for time series classifiers that operates in the decomposition space of Empirical Mode Decomposition. An input signal is split into Intrinsic Mode Functions (IMFs), and selected IMFs are progressively substituted with those of a Nearest Unlike Neighbour (NUN) until the classifier flips to the target class. We evaluate six IMF-selection strategies and a multi-NUN cycling extension on two UCR benchmarks (FaultDetectionA, FruitFlies). The variance-based strategy with three NUNs outperforms two prominent baseline techniques on reliability and plausibility metrics, while cycling across three NUNs yields the best proximity across both datasets.

Udo Schlegel, Julian Rakuschek, Thomas Seidl +3

LMU Munich, Munich, Germany · TU Graz, Institute of Visual Computing, Graz, Austria · Munich Center for Machine Learning (MCML), Munich, Germany +3

Counterfactual Explanation Time-Series Classification Time-Frequency Time Series Dynamic Mode Decomposition Oscillatory Dynamics

Aug 3, 2026q-bio.NC

Detecting high-frequency brain disorder signals using dynamic mode decomposition from EEG

Recent studies have reported clearly identifiable dynamical changes in the high-frequency range of EEG signals recorded during specific stimuli, such as visual or auditory inputs, or in cases of brain disorders like epileptic seizures. In this study, we utilized Dynamic Mode Decomposition (DMD) to extract consistent and persistent dynamical changes in the high-frequency band from the signals of neurologically relevant EEG channels. High-frequency DMD modes were employed as features, composing a feature table. Through post-processing, a random distribution test was performed, revealing that approximately 70% of the samples exhibited consistent high-frequency dynamics within the signal of a specific channel. Furthermore, classification experiments confirmed that the PCA components of the feature table that passed the test formed a consistent pattern that distinguished the alcohol-dependent group from the control group.

Jacob Kang, Jong-Hyeon Seo

Fischell Department of Bioengineering, University of Maryland, College Park, MD, United States · School of Basic Sciences, Hanbat National University, Daejeon, Republic of Korea

Electroencephalography Dynamic Mode Decomposition High-Frequency Reconstruction Principal Component Analysis

Jul 31, 2026cs.LG

Dynamics-aware identification of governing equations from sparse and noisy data

Sparse identification of nonlinear dynamics (SINDy) and PDE functional identification (PDE-FIND) recover parsimonious ordinary and partial differential equations (ODEs and PDEs) from data. However, sparse and noisy temporal measurements can make derivative estimates unreliable. To address this problem, we evaluate Koopman-based upsampling techniques implemented with dynamic mode decomposition (DMD), extended DMD (EDMD), and optimized DMD. These methods learn finite-dimensional approximations of Koopman evolution on selected observables and are used to interpolate and denoise snapshots inside the observed time window before derivative estimation and sparse regression. The empirical benchmark comprises two ODE systems, Lorenz-63 and Van der Pol, and three periodic PDE systems, Burgers, Fisher-Kolmogorov-Petrovskii-Piskunov (Fisher-KPP), and linear advection-diffusion, over sparse and noisy sampling regimes. Polynomial EDMD gives the strongest ODE results, especially in coefficient accuracy. The PDE results are system-dependent: low-rank DMD-assisted reconstructions improve Burgers and advection-diffusion discovery, while the raw baseline (without upsampling) remains competitive for the Fisher-KPP data. A comparison against linear and smoothing-spline interpolation techniques shows that the selected Koopman-based preprocessors provide overall performance gains over these non-dynamical alternatives. We also demonstrate that DMD-assisted upsampling can stabilize Pareto-based non-oracle support-size selection. Overall, Koopman-based upsampling is best viewed as a dynamics-aware preprocessing step that can reduce derivative-estimation error when its observable representation and low-rank structure are appropriate for the data.

Pongpisit Thanasutives, Yoshinobu Kawahara

Center for Advanced Intelligence Project (AIP), RIKEN, Osaka, Japan · 2Graduate School of Information Science and Technology, The University of Osaka, Osaka, Japan

System Identification Dynamic Mode Decomposition Partial Differential Equations One-Dimensional Viscous Burgers Equation Noisy Data Upsampling

Jul 19, 2026cs.AI

Coordinated Disentanglement with Iterative Mode Discovery Under Hidden Correlations

Disentangled representation learning is a powerful paradigm for robust attribute prediction. While recent methods address attribute correlations, hidden correlations remain underexplored, where data under the value of a certain attribute exhibit underlying modes correlated with other attributes. To preserve mode information and achieve disentanglement, we jointly discover modes and enforce mode-based conditional independence. Yet, the interdependency between these two modules may lead to error amplification under naive iterations. We propose Coordinated Disentanglement with Iterative mode Discovery (CoDID), an end-to-end framework featuring a dynamic architecture that adapts to evolving number of modes, and a coordination mechanism that mitigates error amplification via meta-optimization. Empirical results demonstrate the state-of-the-art performance on diverse tasks.

Rong Hu, Ling Chen

State Key Laboratory of Blockchain and Data Security, Zhejiang University, Hangzhou, China · College of Computer Science and Technology, Zhejiang University, Hangzhou, China

Representation Learning Dynamic Mode Decomposition Iterative

Jul 14, 2026cs.LG

Cluster-Weighted EDMD

Extended Dynamic Mode Decomposition (EDMD) approximates Koopman operators from data, but a single global operator is inefficient when different state-space regions exhibit distinct local dynamics. We introduce Cluster-Weighted EDMD (CW-EDMD), which jointly learns a soft phase-space partition and a per-cluster EDMD operator. Its Expectation-Maximization (EM) objective assigns each transition based on both geometric proximity and prediction residuals, so clusters specialize where local Koopman models are accurate rather than where the data are dense. On Lorenz, damped pendulum, and Duffing systems, across 36 configurations and 10 seeds, CW-EDMD improves matched-degree EDMD in one-step and 5s-rollout prediction. Across 288 paired comparisons, there are significant error reductions in 258 cases, increases in 4, and no differences in 26. Median one-step error reductions are 57x, 2.7x, and 12x on pendulum, Duffing, and Lorenz, respectively.

Lorenzo Tomaz, Judd Rosenblatt, Flavio Kicis +2

AE Studio

Dynamic Mode Decomposition Koopman Operator Phase Space Expectation-Maximization

Jul 2, 2026eess.SY

Koopman operator theory: fundamentals, control, and applications

The Koopman operator has gained considerable attention due to its ability to provide a global linear representation of highly complex dynamical systems. The operator describes nonlinear dynamics in a linear way through the lens of real- or complex-valued observable functions. Recently proposed data-driven techniques, like extended dynamic mode decomposition (EDMD), its kernelized variant, and machine-learning methods, can be used to generate finite-dimensional approximations accompanied by finite-data error bounds. In this tutorial paper, we provide a concise introduction into Koopman operator theory and its use in systems and control. A particular focus is put on data-driven surrogate models, their extension to systems with inputs, and controller design using Koopman operator theory. Moreover, we demonstrate the key techniques, i.e., EDMD and Koopman MPC. To this end, we provide simulation studies including source code on GitHub to enable the interested reader to experience the Koopman operator in systems and control step by step.

Igor Mezić, Jorge Cortés, Karl Worthmann +2

Department of Mechanical Engineering, University of California, Santa Barbara, USA · Department of Mechanical and Aerospace Engineering, University of California, San Diego, USA · Optimization-based Control Group, TU Ilmenau, Germany +2

Koopman Operator Dynamic Mode Decomposition Robust Control Model Predictive Control Surrogate Models

Jun 27, 2026math.NA

Residual-Guided Dictionary Learning for Spectrally Accurate Koopman Approximation

Koopman theory promises linear structure in nonlinear dynamics, but numerical Koopman spectra are easy to compute and hard to trust. A finite EDMD matrix always has eigenvalues; the problem is that many of them may have nothing to do with the infinite-dimensional operator. In this paper we make spectral reliability the objective of dictionary learning. We train neural-network dictionaries not merely to predict the next snapshot, but to minimize Residual Dynamic Mode Decomposition residuals: operator-level a posteriori errors that test whether computed eigenvalues and modes are genuine Koopman spectral objects. To keep the learned observables from collapsing into an unstable coordinate system, the loss also penalizes the condition number of the lifted data matrix. Thus the method couples two requirements that should not be separated: small Koopman residuals and a well-conditioned representation. The result is a learned dictionary that is expressive, numerically stable, and spectrally disciplined. Across conservative and dissipative benchmark systems, the method sharply reduces spectral pollution, improves residual pseudospectral inclusion, and lowers forecast error relative to standard fixed dictionaries. On sea-surface temperature data, it gives cleaner Koopman diagnostics and substantially better one-step forecasts from noisy observations with no governing equations. The message is simple: neural Koopman learning should be judged not by prediction alone, but by whether its spectral claims can be certified. Residuals provide the certificate; conditioning makes it computable.

George Coote, Matthew J. Colbrook

Department of Applied Mathematics and Theoretical Physics, University of Cambridge, Wilberforce Road, CB3 0WA, United Kingdom

Koopman Operator Unsupervised Dictionary Learning Dynamic Mode Decomposition Spectral Representation Method Residual Learning Sea-Surface Temperature Data

Jun 18, 2026eess.SY

Topological Data Analysis for High-Dimensional Dynamic Process Monitoring

Real-time process monitoring requires methods that extract actionable information from high-dimensional time-series data. In this work, we present a new approach for process monitoring that combines tools of topological data analysis (TDA) and machine learning. In the proposed approach, we represent multivariate time-series data as manifolds and use topological descriptors to summarize the structure of such data; we then use a neural ordinary differential equation to learn the dynamic evolution of the topological structure of the system. Using real data from an industrial process, we show that this trajectory-based event detection approach is effective at detecting diverse types of events. We contrast this approach against reconstruction-based approaches such as principal component analysis and autoencoders and against a trajectory-based approach that uses Koopman autoencoders.

Angan Mukherjee, Tyler A. Soderstrom, Michael J. Kurtz +1

Department of Chemical & Biological Engineering, University of Wisconsin-Madison, 1415 Engineering Drive, Madison, WI 53706, USA · ExxonMobil Technology and Engineering, 22777 Springwoods Village Pkwy, Spring, TX 77389, USA

Dynamic Mode Decomposition Topology Time Series Autoencoder Architectures

Jun 4, 2026physics.flu-dyn

Multiscale POD of Transformer Attention Fields: Scale-Selective Analysis via Morlet Scalogram

We introduce scale-selective Proper Orthogonal Decomposition (POD) for transformer attention fields, inspired by the use of POD for extracting energetically dominant modes from turbulent flow ensembles. The Morlet continuous wavelet transform identifies dominant temporal scales in the attention lag structure across a document ensemble; POD then extracts the energetically dominant modes at each scale from the ensemble of attention fields. The resulting modes reveal layer-dependent scale organisation, with early layers emphasising fine scales and later layers shifting toward coarser scales. We define a spectral concentration index from the POD eigenvalue decay rate and show empirically that it differentiates layers by their attention field complexity. By the classical POD optimality theorem, the extracted modes minimise the average L2 reconstruction error over the ensemble (Theorem 1), giving a data-driven effective rank for each layer. The method requires no architectural modification and no linguistic annotations: dominant attention patterns emerge from ensemble statistics alone. The turbulence analogy is structural rather than physical: we borrow ensemble covariance and modal analysis, not fluid dynamics itself.

Athanasios Zeris

Independent Researcher, Athens, Greece.

Transformer Attention Dynamic Mode Decomposition

Jun 3, 2026cs.LG

Deep Embedded Multiplicative DMD for Algebra-Preserving Koopman Learning

Koopman theory turns nonlinear dynamics into a linear spectral problem. In computation, however, everything depends on a hard finite-dimensional choice: the observables must be expressive, nearly invariant under the dynamics, and, ideally, compatible with composition. Deep Koopman methods learn flexible coordinates, whereas structure-preserving methods enforce operator identities on fixed dictionaries. We combine these ideas by introducing Deep Embedded Multiplicative Dynamic Mode Decomposition (DeepMDMD), a method that learns a latent space and a partition of it, while enforcing the Koopman product rule as an exact algebraic constraint. Training alternates between an exact multiplicative operator update and a differentiable latent-clustering step that promotes Koopman closure. The result is a finite transition map on learned latent cells. Its nonzero spectrum lies on the unit circle, its dictionary is shaped by the dynamics rather than by ambient geometry, and forecasts are made in latent coordinates before being decoded to physical space. Across Hamiltonian, chaotic, and fluid examples, DeepMDMD learns dictionaries that are far more compact and dynamically coherent than those produced by geometric MDMD partitions. It reduces spectral pollution, reveals richer continuous-spectrum structure, and gives stable forecasts under severe noise. In high-dimensional flows, including a 158,624-dimensional cylinder wake and a noisy

Re=20,000

lid-driven cavity, it preserves coherent structures and long-time spectral statistics where state-space MDMD fails. These results suggest a practical rule for Koopman learning: learn the coordinates, constrain the algebra.

Kelan Gray, Finlay Brown, Nicolas Boullé +1

Department of Mathematics, Imperial College London, London, SW7 2AZ, UK. · Department of Applied Mathematics and Theoretical Physics, University of Cambridge, Cambridge, CB3 0WA, UK.

Koopman Operator Dynamic Mode Decomposition Spectral State Space Models Dataset Distillation

Jun 1, 2026cs.LG

A Systematic Evaluation of Current Architectures in Wind Power Forecasting

Interval wind speed forecasting is essential for the efficient integration of wind energy into power systems, as it accounts for the inherent uncertainty of wind resources. This study presents a systematic literature review focused on hybrid approaches to interval forecasting of wind generation, exploring the combination of deep learning, modal decomposition, and statistical methods. To guide the paper selection, Latent Dirichlet Allocation (LDA) was applied for topic modeling, enabling the identification of patterns and research trends. The findings emphasize that integrating hybrid models with decomposition techniques-such as Variational Mode Decomposition (VMD) and Ensemble Empirical Mode Decomposition (EEMD)-enhances forecast accuracy and reliability by narrowing prediction intervals without compromising coverage. Regarding interval construction, most studies adopt a dual-model strategy, independently forecasting the lower and upper bounds. Input data are commonly decomposed using techniques like EMD, EEMD, or VMD, which extract frequency-based components. These components serve as inputs to models such as LSTM or ELM, trained separately for each bound. This approach allows for targeted modeling of uncertainty, improving flexibility and precision, Interval quality is typically evaluated through metrics that balance coverage and interval width. The review also highlights challenges, including the lack of standardized evaluation metrics, computational complexity, and limited real-world validation. Overall, the study reinforces the value of interval forecasting for wind energy operations and offers insights for advancing model robustness and decision-making.

Vinicius Bortolini, Gilson Adamczuk Oliveira, Erick Oliveira Rodrigues +1

Industrial and Systems Engineering Graduate Program (PPGEPS), Federal University of Technology Parand (UTFPR), Pato Branco, Parand $5503-390, Brazil

Offshore Wind Turbine Wind Dynamic Mode Decomposition

May 14, 2026math.NA

Wavelet-Based Observables for Koopman Analysis: An Extended Dynamic Mode Decomposition Framework

We present an in-depth analysis of the Koopman semigroup via wavelet transform. Towards this goal, we start by introducing the wavelet-based observables and show that they are eigenfunctions of the Koopman semigroup when this semigroup is considered over the Banach space of continuous functions on a compact forward-invariant set endowed with the supremum norm. We then construct closed-form expressions of the action of the Koopman semigroup and its resolvent in terms of these observables. To approximate the action of Koopman semigroup numerically, we combine Extended Dynamic Mode Decomposition (EDMD) with the proposed wavelet-based observables leading to the Wavelet Dynamic Mode Decomposition via Continuous Wavelet Transform (cWDMD) algorithm. We validate our theoretical results on two numerical examples.

Cankat Tilki, Serkan Gugercin

Department of Mathematics, Virginia Tech, Blacksburg, VA - 24061.

Dynamic Mode Decomposition Koopman Operator Wavelets Spectral Decomposition

May 9, 2026stat.ML

Measuring and Decomposing Mode Separation via the Canonical Diffusion

Mode separation, namely how sharply a distribution fragments into barrier-separated clusters, is a fundamental geometric property of densities, difficult to quantify in high dimensions. It is structurally distinct from dispersion, yet existing tools fall short: differential entropy rises with spread regardless of fragmentation, PCA orders directions by variance regardless of barriers, and mutual information requires a mixture decomposition one usually does not have. We measure mode separation through a single stochastic process intrinsic to the density: a unique reversible diffusion with

f

as its stationary distribution and constant scalar diffusion coefficient. We extract two readouts from its autocovariance matrix: SSA (Sum of Squared Autocorrelations), a scalar barrier-sensitive measure; and DA (Dominant Autocorrelation directions), linear projections ordered by metastability rather than variance. Under an isotropic-Gaussian null, we derive a closed-form spectrum for the empirical autocovariance that generalizes Marchenko--Pastur, with an analytic upper edge that selects the lag at which DA is read off. Both readouts use only samples and a score function, scaling to high dimensions through pretrained score-based generative models via Tweedie's identity. We apply our framework to three settings: (i) synthetic Gaussian mixtures, where SSA tracks mutual information; (ii) SDXL text-to-image generations, where SSA and DA capture structure that entropy and PCA miss; and (iii) molecular dynamics of alanine dipeptide, where DA recovers the known slow backbone dihedrals from static samples alone.

Shaul Tolkovsky, Ori Meidler, Or Zuk

Department of Statistics and Data Science Hebrew University of Jerusalem

Dispersion Dynamic Mode Decomposition Covariance Entropy Measures Cluster

May 8, 2026cs.CV

Dynamic Mode Decomposition along Depth in Vision Transformers

Recent work has shown that contiguous vision transformer (ViT) blocks (a) can be replaced by a linear map and (b) organize into recurrent phases of computation. We ask whether these observations coincide: does ViT depth implement approximately \textit{autonomous linear} dynamics, admitting a single operator

K

applied recurrently across a contiguous span? We test this using Dynamic Mode Decomposition (DMD), which fits

K

from selected, consecutive hidden-state pairs and predicts

p

steps ahead via

K^p

. On four pretrained DINO ViTs, we study the regularization, rank, and calibration budget required for stable fitting. For short spans (

p \leq 4

),

K^p

tracks an unconstrained endpoint map to within

0.02

cosine similarity on DINOv3-H/16+, while also recovering intermediate activations at each skipped block. At early cut starts, the fitted operators compress to rank

\ll d

with minimal calibration data, and across tokens, \texttt{cls} is most amenable to linearization; both properties decay monotonically with depth. Yet this local fidelity does not transfer downstream. At the final hidden state, after propagating through the remaining blocks, an identity baseline becomes competitive.

Nishant Suresh Aswani, Saif Eddin Jabari

NYU Tandon & NYU Abu Dhabi

Self-Supervised Vision Transformers Vision Transformer Dynamic Mode Decomposition Transformer Architectures Hidden States Recurrence

May 7, 2026cs.LG

Distributional Spectral Diagnostics for Localizing Grokking Transitions

In grokking, a model first fits the training data while test accuracy remains low, and only later begins to generalize. We ask whether this transition can be localized from observed training trajectories before the test accuracy rises, and formulate grokking transition localization as a diagnostic problem with an explicit threshold/FPR/lead-time trade-off. Task-dependent observables are summarized as empirical distributions, mapped to Wasserstein/quantile coordinates, and analyzed by Hankel dynamic mode decomposition (DMD); the resulting reconstruction residual, together with spectrum and effective rank, forms the diagnostic output. On held-out modular-addition Transformer runs, the residual achieves AUROC

\approx

0.93 for grokking-vs-non-grokking discrimination at the run level; under a fixed sustained-threshold operating rule, true-positive alarms can precede onset, with lead time reported jointly with false-alarm rate and uncertainty intervals. Perturbation experiments show that, in the tested

wd=1

pool, high-residual windows exhibit about

3\times

larger short-horizon perturbation deviation than low-residual windows. In a same-data norm-window control, perturbation sensitivity aligns with the residual ordering rather than total-parameter-norm ordering, suggesting that the residual is not merely a total-norm proxy at the window level in the studied

wd=1

dynamics. Norm signals remain strong run-level regime indicators, and log-probability performs best among the observables tested under the current protocol. We position the residual as a window-level monitoring and localization signal in the studied modular-arithmetic Transformer settings, not a universal early-warning predictor or an intervention rule.

Ziyue Wang, Yufeng Ying, Takafumi Kanamori

Institute of Science Tokyo · University of Science and Technology of China

Grokking Transformer Residual Streams Dynamic Mode Decomposition Spectral Transformer Architectures Diagnosis+1

May 6, 2026cs.CV

Hybrid Congestion Classification Framework Using Flow-Guided Attention and Empirical Mode Decomposition

Accurate traffic congestion classification requires models that jointly capture roadway scene context and non-stationary traffic motion, yet most prior work treats these requirements in isolation. Vision-based methods often depend on appearance cues with standard temporal pooling, which can bias predictions toward static infrastructure, whereas signal-based approaches characterize temporal dynamics but lack the spatial context needed for scene-level localization. These complementary limitations motivate a unified framework that links motion evidence to spatial feature selection while preserving data-adaptive temporal characterization. This study therefore proposes FLO-EMD, a hybrid approach that couples motion-guided attention with empirical, data-driven temporal decomposition. Dense optical flow guides channel and spatial attention so that RGB features are refined toward motion-relevant regions. In parallel, aggregated flow statistics form compact motion traces that are decomposed using Empirical Mode Decomposition (EMD) to extract intrinsic temporal components. The resulting EMD embedding is fused with learned spatiotemporal representations to classify light, medium, and heavy congestion. Experiments on 1,050 five-second clips from four surveillance networks show that FLO-EMD achieves 97.5% overall test accuracy (weighted F1 = 0.9742), outperforming established baselines and remaining robust across diverse environmental conditions; ablation and sensitivity analyses further quantify the contributions of EMD, the number of intrinsic mode functions, and the selected motion descriptors.

Eugene Kofi Okrah Denteh, Blessing Agyei Kyem, Joshua Kofi Asamoah +1

Civil, Construction and Environmental Engineering Department, North Dakota State University, North Dakota, United States of America

Optical Flow Spatiotemporal Representations Dynamic Mode Decomposition

May 3, 2026cs.LG

Learning Koopman operators for coupled systems via information on governing equations of subsystems

Nonlinear coupled systems are ubiquitous in science and engineering. The analysis and modeling of such systems is challenging due to their high dimensionality and complex interactions among subsystems. In recent years, operator-theoretic methods based on the Koopman operator have attracted attention as a powerful tool for analyzing and modeling nonlinear dynamical systems. Extended dynamic mode decomposition (EDMD) is one of the most popular methods to approximate the Koopman operator. However, EDMD is a purely data-driven method, and it could be unstable and inaccurate for coupled systems under limited data availability. In this paper, we propose a method to learn the Koopman operator for coupled systems using the differential equations governing each subsystem. We also demonstrate its effectiveness through numerical experiments on coupled oscillator systems.

Tatsuya Naoi, Jun Ohkubo

Graduate School of Science and Engineering, Saitama University, Sakura, Saitama, 338-8570, Japan

Koopman Operator Dynamic Mode Decomposition Nonlinear Dynamics

May 3, 2026cs.LG

Complex Diffusion Maps with $ω$ -Parameterized Kernels Revealing Inherent Harmonic Representations

In this paper, we propose Complex Diffusion Maps (CDM), a novel diffusion mapping framework that aims to reveal the dominant complex harmonics of high-dimensional data. Inspired by the local Gaussian kernel relevant to the heat equation and the nonlocal Schrödinger kernel relevant to the Schrödinger equation, we propose a unified family of

ω

-parameterized complex-valued kernels for the trade-off between local and nonlocal connections. We establish the theoretical foundation based on the operator spectrum theory, where the corresponding diffusion operator, diffusion distance, and complex harmonic maps are well-defined. An optimization-based interpretation of the maps is also developed, aiming to preserve angular structure in the complex diffusion space rather than relying solely on real-valued magnitude. We extensively evaluate CDM on both synthetic and real-world datasets. The complex-valued kernel amplifies differences among easily confusable samples, improving discriminative power over both linear and nonlinear methods based on real-valued kernels. CDM remains robust in high-noise settings, yielding a clearer eigengap that enhances spectral separation. For resting-state fMRI data, CDM captures more strongly correlated and nonlocal spatiotemporal dynamics. Without task-specific tuning, CDM achieves competitive performance on a public EEG sleep dataset, while maintaining high computational efficiency compared with both traditional machine learning and deep neural network approaches, highlighting its generality and practical value.

Tongzhen Dang, Weiyang Ding, Michael K. Ng

Institute of Science and Technology for Brain-Inspired Intelligence, Fudan University, Shanghai, China. · Institute of Science and Technology for Brain-Inspired Intelligence, Fudan University, Shanghai, China; Key Laboratory of Computational Neuroscience and Brain-Inspired Intelligence (Fudan University), Ministry of Education, China. · Department of Mathematics, Hong Kong Baptist University, Hong Kong, China.

Dynamic Mode Decomposition Kernel Method Fourier Invertible Flows

May 1, 2026eess.SP

Equation-Free Digital Twins for Nonlinear Structural Dynamics

Monitoring high-dimensional engineering structures in extreme environments is limited by non-stationary excitation, nonlinear structural kinematics, and stochastic forcing. Traditional model-based and black-box data-driven methods often struggle to resolve these dynamics in real time, particularly under sensor failure or partial observability. This paper introduces a rank-optimized digital twin framework based on Koopman operator theory, Hankel-matrix embeddings, and dynamic mode decomposition. By lifting operational data into a linear invariant subspace, the method enables autonomous, input-blind reconstruction of structural states without requiring a priori mass or stiffness matrices. The framework is validated on an NREL 5MW spar-buoy floating offshore wind turbine, representing a challenging coupled aero-hydro-servo-elastic system. Results show that the rank-optimized Koopman-Hankel manifold separates structural resonances from deterministic 3P rotor harmonics under colored noise, where standard subspace identification can be unreliable. A rolling-horizon virtual sensing strategy achieves high-fidelity reconstruction at critical structural hotspots, with coefficient of determination greater than 0.95 at 1 Hz data assimilation and accuracy exceeding 0.99 at higher sampling rates. By estimating a physical Lyapunov time of approximately 1.0 s, the study defines the predictability horizon associated with the system information barrier. The proposed framework provides a computationally efficient and resilient digital twin approach for real-time identification and virtual sensing of complex structural dynamics.

Mohammad Mahdi Abaei, Ahmad BahooToroody, Arttu Polojärvi +4

Department of Mechanical Engineering, Marine Technology and Arctic Group, Aalto University, Finland · Independent Expert, Digital Twin Technology, Ribe, Denmark · Institute for Risk and Reliability, Leibniz University Hannover, Germany

Dynamic Mode Decomposition Offshore Wind Turbine Data Assimilation Engineering

Apr 28, 2026math.DS

Dictionary learning for Kernel EDMD

Studying nonlinear dynamical systems through their state space behavior can be challenging, and one possible alternative is to analyze them via their associated Koopman operator. This turns the nonlinear problem into a linear, infinite-dimensional one. To approximate the operator in finite dimensions, extended dynamic mode decomposition (EDMD) is a commonly used algorithm. It requires a finite list of functionals and a set of snapshots from the system to compute an approximation of the operator and its corresponding spectrum. Instead of choosing the list of functionals directly, it can be implicitly defined via kernels, a method known as kernel extended dynamic mode decomposition (kEDMD). However, one still needs to define the kernel and choose its parameter values. In this paper, we aim to streamline this process by extending dictionary learning for EDMD to kernel learning in kEDMD. By simplifying kEDMD we show how to perform gradient-based optimization over the learnable kernel parameters, and demonstrate that this method leads to useful kernels for the original kEDMD. The focus of our work is a method that takes a weighted list of kernels with randomly initialized values as input and outputs a list of kernels and parameter values suitable for approximating the Koopman operator of the underlying system. We demonstrate that unimportant kernels can be removed from the list by analyzing the weights in the weighted sum. We evaluate the method across several experiments, including the Duffing oscillator and the Kuramoto-Sivashinsky PDE, showcasing the method's different strengths.

Erik Lien Bolager, Boumediene Hamzi, Houman Owhadi +2

School of Computation, Information and Technology, Munich Data Science Institute, Munich Center for Machine Learning, Technical University of Munich, Germany · Department of Computing and Mathematical Sciences, Caltech, USA · The Alan Turing Institute, UK +1

Dynamic Mode Decomposition Kernel Method Koopman Operator Oscillatory Dynamics

Apr 24, 2026eess.SP

Modulation Feature Enhancement with a Multi-Stage Attention Network for Underwater Acoustic Target Recognition

Underwater acoustic target recognition is critical for maritime applications, yet it faces challenges arising from the complex and diverse nature of ship-radiated noise. To address these issues, we propose a robust deep learning-based framework. First, we introduce a feature extraction and fusion method based on variational mode decomposition (VMD) and the 3/2-D spectrum to generate high-fidelity 2-D DEMON spectral features, which effectively capture modulation envelope information. To further enhance feature representation, we design a one-dimensional convolutional neural network (1-D CNN) integrated with a novel Multi-Stage Multi-Type Attention Mechanism (MMATT) that adaptively refines features at different network depths. Within this mechanism, we propose a Residual Channel-Independent Spectral Attention Mechanism (R-CISAM) and a Multi-Scale Separate-and-Fuse Spectral Attention Mechanism (MS-SFSAM). Moreover, to mitigate performance degradation caused by severe class imbalance inherent in real-world ship-radiated noise data, we devise an Adjustable Class-Balanced Focal Loss (ACBFL), which provides flexibility across tasks with varying degrees of imbalance. Experimental results on a real-world ship-radiated noise dataset demonstrate that the proposed solutions effectively enhance underwater acoustic target recognition performance.

Jiaping Yu, Shefeng Yan, Linlin Mao +2

Underwater Acoustic Target Recognition Dynamic Mode Decomposition

Dec 18, 2025eess.SP

Few-Shot Specific Emitter Identification via Integrated Complex Variational Mode Decomposition and Spatial Attention Transfer

Specific emitter identification (SEI) utilizes passive hardware characteristics to authenticate transmitters, providing a robust physical-layer security solution. However, most deep-learning-based methods rely on extensive data or require prior information, which poses challenges in real-world scenarios with limited labeled data. We propose an integrated complex variational mode decomposition algorithm that decomposes and reconstructs complex-valued signals to approximate the original transmitted signals, thereby enabling more accurate feature extraction. We further utilize a temporal convolutional network to effectively model the sequential signal characteristics, and introduce a spatial attention mechanism to adaptively weight informative signal segments, significantly enhancing identification performance. Additionally, the branch network allows leveraging pre-trained weights from other data while reducing the need for auxiliary datasets. Ablation experiments on the simulated data demonstrate the effectiveness of each component of the model. An accuracy comparison on a public dataset reveals that our method achieves 96% accuracy using only 10 symbols without requiring any prior knowledge.

Chenyu Zhu, Zeyang Li, Ziyi Xie +1

School of Information Science and Technology, Harbin Institute of Technology (Shenzhen), Shenzhen 518055, China · School of Electrical Engineering and Computer Science, and the Digital Futures, KTH Royal Institute of Technology, 114 28 Stockholm, Sweden · Ranplan Wireless Network Design Ltd., Cambridge, CB23 3UY, U.K.

Spatial Attention Dynamic Mode Decomposition

Jul 16, 2025cs.LG

HyDRA: A Hybrid Dual-Mode Network for Closed- and Open-Set RFFI with Optimized VMD

Device recognition is vital for security in wireless communication systems, particularly for applications like access control. Radio Frequency Fingerprint Identification (RFFI) offers a non-cryptographic solution by exploiting hardware-induced signal distortions. This paper proposes HyDRA, a Hybrid Dual-mode RF Architecture that integrates an optimized Variational Mode Decomposition (VMD) with a novel architecture based on the fusion of Convolutional Neural Networks (CNNs), Transformers, and Mamba components, designed to support both closed-set and open-set classification tasks. The optimized VMD enhances preprocessing efficiency and classification accuracy by fixing center frequencies and using closed-form solutions. HyDRA employs the Transformer Dynamic Sequence Encoder (TDSE) for global dependency modeling and the Mamba Linear Flow Encoder (MLFE) for linear-complexity processing, adapting to varying conditions. Evaluation on public datasets demonstrates state-of-the-art (SOTA) accuracy in closed-set scenarios and robust performance in our proposed open-set classification method, effectively identifying unauthorized devices. Deployed on NVIDIA Jetson Xavier NX, HyDRA achieves millisecond-level inference speed with low power consumption, providing a practical solution for real-time wireless authentication in real-world environments. The source code is published on https://github.com/Crazy-Bull/HyDRA.

Hanwen Liu, Yuhe Huang, Yifeng Gong +2

School of Electronics and Communication Engineering, Sun Yat-sen University, Shenzhen, 518107, China

Fingerprint Dual-Path Network Random Forest Dynamic Mode Decomposition