This study proposes an end-to-end architecture that integrates a learnable signal decomposition module with both frequency-independent and frequency-dependent Koopman based networks for sequence forecasting and demonstrates that decomposing a signal into a frequency-independent trend component and a frequency-dependent periodic component improves prediction accuracy when perfect linearization is unattainable.
Abstract
Koopman theory offers a linear-operator view of nonlinear sequence dynamics by lifting observations into a space where evolution is governed by a linear time-invariant Koopman operator. While the Koopman operator provides a linear representation of nonlinear dynamics, it is generally infinite dimensional and defined under time-invariant assumptions. To model non-stationary signals with frequency-dependent behavior, a frequency-varying extension is required. In recent years, deep learning has been increasingly employed to exploit its powerful function-approximation ability for learning the Koopman operator. In this study, we propose a novel approach called neural decomposition Koopman (NDKoop), an end-to-end architecture that integrates a learnable signal decomposition module with both frequency-independent and frequency-dependent Koopman based networks for sequence forecasting. To the best of our knowledge, this is the first work to jointly realize end-to end Koopman modeling and signal decomposition within a unified neural framework. We demonstrate that decomposing a signal into a frequency-independent trend component and a frequency-dependent periodic component, each governed by a corresponding Koopman operator, improves prediction accuracy when perfect linearization is unattainable. Numerical experiments across several forecasting benchmarks indicate that the proposed NDKoop provides strong performance.
One of the main objectives in control theory is to obtain a linear representation of inherently nonlinear systems in order to leverage the analytical and theoretical tools developed for linear systems. In this context, the Koopman operator has attracted increasing interest in recent years.Koopman operator theory provides a framework in which nonlinear dynamical systems are represented by a linear operator acting on an infinite-dimensional Hilbert space. Since such an infinite-dimensional representation is not numerically tractable, numerous finite-dimensional approximation methods have been proposed. These approaches typically rely on time-series data and include extended dynamic mode decomposition as well as deep learning–based variants. In this paper, we propose an original machine-learning-based approach for the synthesis of a fixed-dimensional Koopman approximant (lifting) of continuous-time nonlinear systems. A differential state-space representation of the system (as opposed to a recurrent state model) is assumed to be available through its vector field (f). The proposed encoder departs from conventional approaches in that it does not directly output the current latent state, but instead generates samples of the latent trajectory evaluated at user-defined time instants (temporal discretization). This formulation enables the integration into the learning process of Physical & Latent Continuous Losses, enforcing consistency between the physical dynamics and the Koopman dynamics, as well as Physical & Latent Boundary Losses, ensuring consistency with the prescribed initial conditions. In parallel, we introduce a structural stability constraint on the Koopman operator. The effectiveness of the proposed methodology is demonstrated through the analysis and simulation of two polynomial dynamical systems.
M. Zodros, A. Colotti, M. Yagoubi et al.· International Conference on...· 0 citations
This work first learns an implicit spectral predictor using Observation Spectral Filtering using Observation Spectral Filtering, a convex method that competes with the best linear observer for the system, and applies spectral-to-LDS distillation to convert this predictor into an explicit recurrent linear dynamical system.
Liane Galanti, Devan Shah, Shlomo Fortgang et al.· 0 citations
Multivariate time series forecasting that uses past observations to predict future data has been widely used in various application. Numerous methods have been developed, including deep learning based approaches. One of the most successful models is Mamba, which is based the state space models (SSMs). However, Mamba and its variants still face the challenges of long training times owing to its nonlinear operations. In this paper, we proposed a Fast Exponential Decay Approach (FEDA) to improve the training efficiency of Mamba. We have observed that the hidden states in the model are linear combinations of past inputs. Therefore, we attempted to use a fully connected layer to generate all the hidden states. Although a fully connected layer does not have a recurrent structure, based on the intuitive idea in multivariate time series forecasting that the influence of observations diminishes over time, we multiplied the weights of the fully connected layer by a time-dependent coefficient to emulate a recurrence-like behavior. We evaluated our model on various datasets, and the results show that our model achieves the performance comparable to the state of the art models, while significantly improving their training efficiency. More specifically, it reduces training time by 20% to 80% depending on the channels of the dataset.
Long-Term Time Series Forecasting (LTSF) is pivotal in domains like energy and traffic management but necessitates capturing intricate dependencies over extended windows. While Transformer-based models dominate, they suffer from quadratic complexity and positional insensitivity. Conversely, recent lightweight MLP/RNN-based models often forcibly compress conflicting dynamic features—linear trends and non-linear fluctuations—into a single channel, leading to suboptimal accuracy. To address these limitations, we propose the Decomposed Recurrent Neural Network (DeRNN). Our approach decouples global trend modeling from local fluctuation extraction via an asymmetric dual-track architecture. Specifically, we introduce a Trend Anchor Track to preserve global scale via direct linear projection, and a Seasonal Feature Track utilizing Bi-directional GRUs to capture complex non-linear dependencies within a reversible normalized space. Extensive experiments on seven benchmarks demonstrate that DeRNN achieves highly competitive, and in most cases superior, accuracy against state-of-the-art methods while maintaining extremely low latency and memory usage. Furthermore, the model exhibits superior robustness against noise and distribution shifts.
Shanyun Qian· Poster Volume 0008 The 2026...· 0 citations
Forecasting time series over long horizons is essential for proactive decision-making in many systems. Recent research has focused on transformer-based architectures, which capture long-range dependencies in sequential data. However, several studies show that simpler linear models can outperform transformers by avoiding overfitting during training. In this context, we present NeuroFlexMLP, a deep learning model for multivariate time series forecasting tasks. NeuroFlexMLP's key distinct feature is the adaptability to the diverse complexity of real-world time series, which is achieved, from the architecture standpoint, by adding non-linear residual blocks to a first linear block. This architectural design simplifies hyperparameter optimization, leading to accurate forecasts for various time series data types regardless of the lookback or prediction horizons, outperforming state-of-the-art (SOTA) models on challenging real-world datasets. Its Multi-Layer Perceptron (MLP) design ensures high computational efficiency, making it scalable for longer input sequences than transformer-based models. We validate NeuroFlexMLP for the LEO satellite beam hopping use case, where its lightweight design enables on-board deployment, and on state-of-the art AI datasets. Across all these benchmarks, NeuroFlexMLP achieves competitive accuracy over state-of-the-art models while providing an adaptive architecture that significantly reduces computational overhead. On the LEO beam hopping task, it achieves up to 35.9% MSE reduction over Informer, which translates into up to 28% lower provisioning cost under asymmetric cost models that penalize under-allocation more heavily than over-allocation.
P. F. Pérez, Claudio Fiandrino, Marco Fiore et al.· La Main· 0 citations
This paper proposes a novel data-driven algorithm to approximate the dominant eigenfunctions (aka.~modes) of the Koopman operator of nonlinear dynamical systems using neural networks. The relevance of learning the dominant Koopman modes is to approximate nonlinear dynamics by linear ones in a lifted space, thereby enabling simplified control and analysis. To fight the curse of dimensionality arising from using expressive templates (here neural networks) for the mode approximation, the proposed method leverages a power-iteration scheme that directly learns the dominant Koopman modes without explicitly constructing the projection of the Koopman operator on the template of functions. Our approach connects to other approaches in the literature that avoid the curse of dimensionality by learning small dictionaries of functions, but differs from them in that we do not require ``anti-collapse mechanisms''to ensure that the learned dictionary is expressive enough to approximate the Koopman operator since our power-iteration scheme is designed to converge toward the dominant modes of the projected Koopman operator. The approach is fully data-driven, requiring only sampled state transitions. Theoretical guarantees are provided, showing convergence under increasing sample size and network width (in connection with the neural tangent kernel theorem). Numerical experiments demonstrate that the method achieves accurate and smooth approximations of dominant modes while avoiding the limitations of traditional techniques such as extended dynamic mode decomposition.
Guillaume O. Berger, Raphael M. Jungers· 0 citations