Jul 2026· International Journal of Robust and Nonlinear Control· 0 citations· 36 references
TL;DR
A novel Koopman operator‐based framework that integrates deep neural networks with subspace identification that achieves high modeling accuracy with significantly reduced errors is proposed, offering an effective and robust solution for nonlinear system modeling.
Abstract
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.
This study aims to compare the effectiveness of the sub-equation neural network method (SENNM) and the neural network predictions method (NNPM) for solving nonlinear systems commonly encountered in fluid dynamics and nuclear physics. The comparison focuses on accuracy, convergence behavior, training performance, and the preservation of key physical properties. The SENNM integrates analytical sub-equation structures with neural network learning to efficiently capture localized waveforms and soliton-like behaviors. In contrast, the NNPM employs a fully data-driven or physics-informed neural architecture to approximate global solutions without relying on predefined functional assumptions. Both methods are implemented on representative nonlinear differential systems and evaluated using numerical simulations, training-loss analysis, and prediction-performance metrics. Numerical results demonstrate that SENNM achieves higher accuracy in modeling systems with strong nonlinear interactions and supports stable learning dynamics. NNPM exhibits superior robustness and generalization across varying boundary and initial conditions. A detailed comparison of neural network predictions and training losses shows that SENNM converges faster with lower residual error for localized structures, while NNPM maintains consistent prediction quality across the full domain. Figures depicting neural network predictions and training-loss curves further illustrate the relative performance, highlighting SENNM’s sharper solution reconstruction and NNPM’s smoother global approximation behavior. The findings reveal that SENNM and NNPM provide complementary advantages for solving nonlinear physical systems. SENNM excels in accuracy and convergence for highly nonlinear or localized patterns, whereas NNPM offers broader predictive stability and adaptability. Together, these insights support the advancement of hybrid analytical–neural and prediction-based neural frameworks for modeling complex fluid and nuclear physics phenomena.
Mustafa Kaytan, Cihan Tiken, Harun Çiçek et al.· Journal of King Saud Univers...· 0 citations
This paper introduces Hybrid Physics-Augmented Neural Network (HyPA-Net), a hybrid modeling framework that integrates physics-based linear time-invariant models with artificial neural networks (ANNs) to address dynamic system modeling when only partial physical knowledge is available. The approach leverages the interpretability and robustness of established physical models while using ANNs—such as long short-term memory architectures—to capture unknown or nonlinear system behaviors. The methodology normalizes state and input variables for compatibility with ANN training and expands traditional recursive state-space equations for efficient backpropagation over sequences. Vehicle dynamics, specifically using a rear-wheel steering test case, validate the proposed framework. Various HyPA-Net configurations are benchmarked against pure physics-based and pure data-driven models, demonstrating improved prediction accuracy and model flexibility. The experimental results in this application confirm that hybrid models yield superior performance over strict physical approaches and can implicitly approximate submodel dynamics within a unified, yet modular, architecture, opening avenues for applications in domains where partial physics-based knowledge is available but insufficient on its own.
Laurin Ludmann, Jaeyoun Choi, J. Neubeck et al.· Vehicles· 0 citations
Real-time control of multivariable nonlinear processes requires models balancing high fidelity with computational tractability. This paper compares two data-driven paradigms: Bilinear Koopman Realizations and Physics-Informed Neural Networks. While standard Koopman approaches seek global linearization, we leverage a bilinear framework in the lifted functional space to preserve the natural coupling of control-affine systems. Simultaneously, PINNs ensure physical consistency by embedding conservation laws into the learning objective. To facilitate high-performance control, both surrogate models are integrated into a nonlinear model predictive control scheme using the CasADi framework, enabling efficient algorithmic differentiation for optimization. Simulation results for a quadruple tank system demonstrate that both paradigms reach mean VAF values above 99%, but the Bilinear Koopman model delivers a lower mean RMSE during step transients while doubling the computational speed of the PINN with a Real-Time Factor above 22. We conclude that despite structural scaling limitations regarding neural exploding gradients and operator instability risks, the Bilinear Koopman realization provides a more reliable, noise-resilient solution for real-time hydraulic benchmarks.
Amir Vanegas, Julio Barón-Velandia, Nelson Leonardo Díaz-Aldana et al.· International Conference on...· 0 citations
High-fidelity parametric simulations of nonlinear magnetodynamic systems are computationally demanding, making real-time and many-query applications impractical with conventional finite element methods (FEMs). While model order reduction (MOR) techniques address this challenge, extending them to nonlinear, time-dependent problems remains difficult, particularly when reduced-order differential equations must still be solved in the latent space. This article proposes a fully data-driven, nonintrusive MOR framework that combines a parametric autoencoder (AE) with a harmonic forecaster neural network, eliminating the need to solve any reduced-order dynamical equations. The AE compresses high-dimensional magnetodynamic field snapshots into a 4-D latent space, while the harmonic forecaster directly predicts the temporal evolution of latent trajectories by decomposing them into harmonic and learnable residual components. The framework is validated on a nonlinear inductor problem with three excitation waveforms, sinusoidal, triangular, and pulsewidth modulation (PWM), and assessed using a domain-anchored fivefold cross-validation strategy over a broad range of current amplitudes and frequencies. For sinusoidal and triangular excitations, the reduced-order model achieves mean field reconstruction errors of 0.68% and 0.80%, respectively, with mean flux linkage errors below 1%. For the significantly more challenging PWM case, mean field and flux linkage errors remain below 3.18%, demonstrating robustness under highly harmonic-rich excitations.
Sina Toghranegar, H. Kazmi, G. Deconinck et al.· IEEE transactions on magneti...· 0 citations
Deep learning has proven highly effective for nonlinear system identification, but heavily parameterized neural networks are prone to overfitting in low-data regimes and lack reliable uncertainty quantification. The recently developed manifold meta-learning framework addresses the data efficiency problem by restricting the model parameters to a meta-learned low-dimensional manifold. However, that method is purely deterministic. We propose a fully probabilistic extension of the manifold meta-learning framework, based on amortized Variational Inference, where a generative prior over the low-dimensional parameter manifold is learned. During task-specific adaptation, we combine Maximum A Posteriori estimation with the Laplace approximation to yield a mathematically grounded posterior approximation. Evaluated on a static regression task and the Bouc--Wen dynamical system benchmark, the proposed approach achieves predictive accuracy comparable to its deterministic counterpart while successfully providing calibrated uncertainty bounds in severely low-data regimes.
A novel nonlinear DDPC framework via a structured OP and a kernelized innovation‐based feedback mechanism is proposed, which yields lower prediction errors and better tracking performance than the existing linear and nonlinear DDPC methods.
Yibo Wang, Yunxiang Ma, Tao Liu et al.· International Journal of Rob...· 0 citations