reduced-order modeling

This work addresses the reduced robustness of projection-based reduced-order models (ROMs)—such as Proper Orthogonal Decomposition (POD) and Neural Ordinary Differential Equations (Neural ODEs)—in high-dimensional dynamical systems under input data perturbations. We propose a novel training framework that synergistically integrates variational data assimilation (VDA) with supervised learning. Crucially, the VDA mechanism is embedded into the ROM training pipeline to enable dynamic correction of input disturbances, thereby significantly enhancing model stability and prediction accuracy under noisy conditions. The framework is model-agnostic and seamlessly accommodates diverse ROM paradigms, including POD and Neural ODEs. Experimental evaluation on graph-structured dynamical systems demonstrates a 42% reduction in mean prediction error under perturbations. Cross-model generalization tests further confirm consistent and robust performance improvements across different ROM architectures.

To address the dual demand for low computational overhead and high-fidelity dynamic models in real-time nonlinear optimization and model predictive control (MPC) for process engineering, this paper systematically reviews and empirically compares eight classes of nonlinear model order reduction (MOR) methods. We propose a novel manifold-Galerkin extension framework tailored to input-driven dynamical systems—marking the first generalization of manifold-Galerkin methods to controlled nonlinear systems. A unified theoretical analysis is provided, characterizing foundational principles, applicability boundaries, and accuracy-efficiency-robustness trade-offs across both generic and process-specific MOR techniques. Quantitative evaluation is conducted on a high-fidelity air separation unit model, assessing methods along four dimensions: approximation accuracy, computational efficiency, robustness to operating condition shifts, and interpretability. The results yield a practical, application-oriented MOR method selection guideline for industrial dynamic modeling.

This work addresses the challenge of constructing efficient surrogate models for parametrized systems in multi-query scenarios—such as optimization, control, and uncertainty quantification—by proposing a unified scientific machine learning framework that systematically integrates physics-driven, data-driven, and hybrid modeling paradigms. The framework encompasses techniques including Proper Orthogonal Decomposition (POD), Proper Generalized Decomposition (PGD), and neural networks, viewed through the lens of function approximation. A key innovation lies in unifying the selection of reduced-order bases and approximation criteria within a coherent analytical framework. Furthermore, the study explores emerging directions such as multi-fidelity fusion, adaptive sampling, and data augmentation. The resulting methodology offers both theoretical foundations and novel modeling paradigms with broad applicability in digital twins, smart manufacturing, and personalized medicine.

Traditional nonlinear model order reduction (MOR) methods struggle to simultaneously achieve high accuracy and efficiency for wave propagation and transport-dominated problems, due to slow decay of the Kolmogorov $n$-width. Method: This paper proposes a novel MOR framework based on temporal-domain expansion and linear encoding. It leverages joint expansion of system dynamics and temporal information to replace the nonlinear encoder in autoencoders with a simple linear projection—enabling high-fidelity state compression without nonlinearity. Contribution/Results: We provide the first theoretical proof that such linear encoding suffices under temporal expansion. The method reduces the number of tunable hyperparameters by ~50%, effectively mitigating the Kolmogorov barrier. It significantly lowers training time and computational cost while preserving approximation accuracy. Experiments demonstrate strong generalization, ease of training, and good scalability—establishing a practical new paradigm for real-time simulation and control of high-dimensional nonlinear dynamical systems.

In parametric dynamical systems, the Proper Orthogonal Decomposition (POD) basis drifts with parameters, degrading the accuracy of reduced-order models (ROMs). Method: This paper proposes the Projected Gaussian Process (pGP) framework—the first to formulate subspace adaptation as a statistical learning task mapping parameter space to the Grassmann manifold. It employs a two-stage geometric mapping: Euclidean space → horizontal space → Grassmann manifold, integrating POD, exponential/logarithmic maps, horizontal-space projection, and Gaussian process regression to enable uncertainty-aware POD subspace prediction while preserving manifold structure. Contribution/Results: Numerical experiments demonstrate that pGP significantly improves ROM accuracy and robustness in both parametric extrapolation and interpolation scenarios, and provides interpretable, calibrated confidence quantification—establishing a new paradigm for parameter-sensitive model reduction.

Existing parametric dynamic mode decomposition (DMD) methods suffer from unstable predictions and limited accuracy under sparse data or in high-dimensional parameter spaces. This work proposes parametric interpolation DMD (piDMD), which, for the first time, directly embeds parametric affine structure into the DMD regression process to construct a unified Koopman-based reduced-order model. The resulting framework enables efficient prediction of system dynamics at unseen parameter values without retraining. By circumventing post-hoc interpolation of modes, eigenvalues, or operators, piDMD significantly enhances robustness and generalization in multidimensional parameter spaces. Demonstrated on nonlinear systems—including flow past a cylinder, electron beam oscillations, and virtual cathode oscillations—the method achieves high-fidelity long-term predictions with only a few training samples, outperforming current parametric DMD approaches.

This work addresses the challenges of modeling complex continuous dynamical systems—particularly strong nonlinearity, high-dimensional state spaces, and difficulties in uncertainty quantification—by proposing a novel framework that integrates Gaussian process ordinary differential equations with second-order reduced-order modeling. The approach learns latent-space dynamics through kernel-based autonomous ODEs and employs spherical projection to ensure numerical stability. Theoretically grounded with convergence guarantees, the method achieves substantially improved prediction accuracy and computational efficiency. Empirical evaluations on multiple benchmark systems demonstrate its superior performance over mainstream techniques such as extended dynamic mode decomposition, exhibiting greater robustness and practicality in terms of predictive accuracy, computational cost, and uncertainty quantification.

This work addresses the challenge of applying high-dimensional dynamical systems to nonlinear control design by proposing an end-to-end joint training framework that yields low-dimensional, control-affine reduced-order models. The approach employs an autoencoder to map high-dimensional states (and inputs) into a latent space, where a state-space model preserving the control-affine structure is learned. A sequential modeling mechanism is incorporated to integrate historical information, thereby enhancing prediction accuracy. Notably, this method achieves the first joint optimization of an autoencoder with a structurally constrained reduced-order model, enabling advanced control strategies such as feedback linearization. In two numerical case studies, the proposed model significantly outperforms linear dynamic baselines in both predictive accuracy and trajectory tracking performance.

This work addresses the problem of constructing linear reduced-order models from sampled data of a system’s transfer function and its derivatives while preserving essential structural properties. The authors propose a novel approach that integrates symmetric Hermite quadrature with balanced truncation. By leveraging sampled information of the transfer function and its derivatives, and incorporating a symmetric Hermite quadrature formula, the method rigorously preserves key characteristics of the original system—such as state-space Hermiticity and asymptotic stability—throughout the reduction process. The resulting reduced-order models not only achieve high-fidelity approximation of the full-order system’s dynamic response but also guarantee stability and physical consistency, thereby significantly enhancing the structure-preserving capability of data-driven model reduction.

This work addresses the instability of conventional machine learning–based reduced-order models for stiff dynamical systems under explicit integration, a challenge exacerbated by the high computational cost and low training efficiency of implicit methods. The authors propose Trajectory-Optimized Time Reparameterization (TOTR), which formulates time remapping as an arc-length coordinate optimization problem aimed at maximizing trajectory smoothness. By minimizing the acceleration of the reparameterized trajectory, TOTR substantially enhances its learnability while enabling efficient explicit integration. Evaluated on three classes of stiff systems, the method achieves training losses one to two orders of magnitude lower than existing benchmarks and significantly improves prediction accuracy in physical time.