dynamical low-rank approximation

提出了一种新的在线自适应非侵入式降阶建模策略，通过流形插值和子空间更新来加速流固耦合问题的收敛，无需存储高维数据。

This work addresses the significant accuracy degradation of traditional reduced-order models (ROMs) when online dynamics deviate from the offline training regime. To overcome this limitation, the authors propose an adaptive projection-based ROM framework leveraging incremental singular value decomposition (iSVD). The approach incorporates occasional full-order evaluations to acquire correction snapshots, enabling online updates of the reduced basis while simultaneously refining both the reduced operators and the hyper-reduction mechanism. A novel history-aware iSVD strategy is introduced to effectively preserve essential system evolution information, markedly outperforming instantaneous update methods. Numerical experiments on nonlinear dynamical systems—including the Burgers equation, Sod shock tube, and a rotating detonation engine—demonstrate that the proposed method surpasses existing adaptive ROM benchmarks in both predictive accuracy and computational efficiency, with negligible overhead from iSVD updates.

Existing reduced-order models (ROMs) for high-dimensional time-varying PDE systems suffer from poor generalizability and prediction drift due to inconsistencies between learned latent dynamics and discrete physical constraints. Method: This paper proposes a Physics-Embedded Differentiable ROM, integrating a differentiable PDE solver (implemented in JAX) as a hard constraint within the latent neural dynamics training pipeline—thereby enforcing adherence to discretized physical laws during learning. The approach jointly combines parameterized manifold learning, physics-informed loss design, and data assimilation under sparse observations. Results: Evaluated on multiple PDE benchmarks, the method substantially outperforms state-of-the-art data-driven and physics-informed ROMs: it enables robust cross-parameter extrapolation, long-term stable predictions beyond training time horizons, accurate modeling with minimal training data, and high-fidelity field reconstruction from sparse, irregularly sampled observations.

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 paper addresses the challenges of identifying recursive structures and ensuring interpretability in modeling dynamic financial processes. We propose a structured reservoir modeling approach that integrates nonlinear time-delay embedding with sparse regression. Our key contribution is the first incorporation of reservoir computing into an interpretable regression framework, enabling joint optimization of sparse least-squares estimation and structured matrix approximation to explicitly characterize system-level recursive dynamics. The method achieves high-accuracy structural identification and long-horizon forecasting across diverse financial time series. It demonstrates robustness to both chaotic and non-chaotic dynamics, significantly enhancing model transparency, generalizability, and predictive reliability compared to conventional black-box reservoir models.

This work addresses the degradation in accuracy of conventional reduced-order models when online dynamics deviate from the training distribution, a limitation stemming from their reliance on external information to update the reduced subspace. The authors propose an intrinsic span-learning mechanism that, for the first time, reveals endogenous signals embedded within the model’s own trajectory, which can be leveraged for adaptation. By employing incremental singular value decomposition with forgetting, the method dynamically reweights and aligns the subspace basis, recasting basis reconstruction as a dynamic preconditioner from the perspective of dynamical systems. This enables in-context learning without external supervision. The approach demonstrates significantly enhanced adaptability and predictive accuracy in out-of-distribution scenarios across three benchmark problems: three-dimensional helical flow, the viscous Burgers equation, and Fisher–KPP dynamics.

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 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 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.

This work addresses the challenge of modeling high-dimensional nonlinear physical systems in the absence of explicit governing equations by proposing a novel approach that integrates neural implicit fields with spectral decomposition of the Koopman operator. By factorizing and decoupling spatial modes from temporal dynamics, the method constructs a generalizable, parameterized flow operator capable of explicitly learning the system’s spectral structure without requiring prior knowledge of the underlying dynamics. It represents the first integration of neural implicit representations with dynamic mode decomposition, enabling stable long-term prediction, interpolation across parameters, and accurate identification of eigenmodes, eigenvalues, and stability characteristics. The framework demonstrates high accuracy and strong generalization across diverse spatiotemporal dynamical systems.