Jul 2026· Journal of Guidance Control and Dynamics· 0 citations· 26 references
TL;DR
The weak graph Koopman bilinear form model is proposed, which integrates geometric deep learning and Koopman theory to learn latent-space dynamics for networked systems, especially for challenging cases that have multiple timescales.
Abstract
This paper presents a framework for developing data-driven, control-oriented models of networked systems, i.e., systems that involve many interacting dynamic components. First, a general formulation named the weak latent dynamics model (wLDM) is developed for learning generic nonlinear dynamics with control. Leveraging the weak form, the wLDM enables more numerically stable and computationally efficient training, as well as more accurate prediction when compared to conventional methods such as neural ordinary differential equations. Building upon the wLDM, we propose the weak graph Koopman bilinear form model, which integrates geometric deep learning and Koopman theory to learn latent-space dynamics for networked systems, especially for challenging cases that have multiple timescales. The proposed methods are demonstrated on three examples of increasing complexity, from academic problems to an application of an electrified aircraft energy system, showing that they achieve superior predictive accuracy and training efficiency when compared to baseline models. Parametric studies provide insights into the effects of hyperparameters in the weak form. The proposed framework shows the capability to efficiently capture control-dependent dynamics in these systems, including stiff dynamics and multiphysics interactions, offering a promising direction for learning control-oriented models of complex networked systems.
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
Time series in real-world applications are often generated by nonlinear dynamical systems, making accurate forecasting challenging. Existing approaches that explicitly model system dynamics typically rely on linear assumptions or Koopman-based linearizations, which may inadequately capture complex nonlinear behaviors and lead to error accumulation in long-horizon prediction. To address this limitation, we propose the Neural Bilinear Dynamical Model (NBDM), which models nonlinear system dynamics through a bilinear latent dynamical formulation. Specifically, NBDM leverages Koopman theory to lift the original nonlinear dynamics into a higher-dimensional latent space, where a bilinear dynamical model is constructed to characterize state evolution. To mitigate the approximation error introduced by bilinear representations, we further incorporate a parameterized error compensation term. Within this formulation, control inputs are explicitly integrated into the dynamics, using auxiliary variables when available and learned feedback signals otherwise. To handle scenarios with missing control inputs, we design a memory-enhanced controller that infers latent controls through multiplicative interactions between historical states and control signals. Experiments on five real-world datasets demonstrate that NBDM consistently outperforms competitive baselines in both given-control and missing-control settings, particularly for multi-step and long-horizon forecasting.
Mengzhou Gao, Huangqian Yu, Pengfei Jiao· Proceedings of the 32nd ACM...· 0 citations
Traditional methods for modelling nonlinear dynamic systems often suffer from high computational complexity and limited prediction accuracy. To address these challenges, this paper proposes a novel Koopman operator‐based framework that integrates deep neural networks with subspace identification. The core idea of the framework is to decouple the overall modeling task by assigning distinct roles to its components: the deep neural network is dedicated to learning the nonlinear lifting map, while subspace identification handles the linear dynamic evolution in the lifted space. This approach improves computational efficiency while preserving model interpretability. Comparative experiments on both weakly nonlinear and strongly nonlinear systems show that the proposed method achieves high modeling accuracy with significantly reduced errors. Further theoretical stability analysis and validation under parameter perturbations confirm the reliability of the proposed approach, offering an effective and robust solution for nonlinear system modeling.
Zixiang Yuan, Jie Ding, Dezhi Shen et al.· International Journal of Rob...· 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
A physics-informed Koopman representation based on generalized momenta is introduced, yielding a linear control-affine model in lifted coordinates with known input structure that avoids the bilinear state – input coupling inherent in standard Koopman approaches, enabling improved prediction accuracy and tractable controller synthesis.
Learning continuous-time representations of dynamical systems from observation data has emerged as a cornerstone of data-driven control and scientific machine learning. However, existing neural differential equations either treat external control inputs heuristically without providing strict structural guarantees, or enforce stability properties under the restrictive assumption of constant or vanishing inputs. This paper proposes the Input-Contraction Neural Differential Model (ICNDM), a novel deep learning framework that seamlessly incorporates time-varying control inputs while ensuring incremental exponential convergence via input-dependent contraction regularization. By leveraging an embedded input encoder and a parameterized metric network, the proposed architecture learns both the non-autonomous neural vector fields and a generalized Riemannian contraction metric simultaneously. We derive sufficient conditions for input-dependent contraction and formally establish an input-to-state contraction property under bounded external excitations. Extensive numerical evaluations on highly nonlinear chaotic oscillators and experimental data from a Permanent Magnet Synchronous Motor (PMSM) drive system demonstrate that ICNDM yields substantial reductions in long-horizon rollout errors and exhibits superior structural robustness against input perturbations compared with state-of-the-art neural differential benchmarks.