linear stability analysis

Traditional linearization-based analyses often fail to accurately characterize the stability of optimization algorithms under nonlinear dynamics, leading to potential misjudgments. This work addresses this limitation by explicitly analyzing the nonlinear behavior of gradient descent (GD) and stochastic gradient descent (SGD) near minima. It proposes a multivariate GD stability criterion based on higher-order derivatives that captures stable oscillatory regimes beyond the reach of linear analysis. Furthermore, the study reveals that the overall stability of SGD can be dominated by a single unstable batch rather than governed by averaging effects. Theoretically, it is proven that when all batches are linearly stable, SGD is nonlinearly stable in expectation; however, the presence of even one unstable batch is sufficient to induce global divergence.

This work addresses the lack of theoretical guarantees for the reliability of Koopman eigenpairs computed from noisy data in data-driven spectral analysis. It introduces, for the first time, shadowing trajectory theory combined with backward error analysis to interpret the residual of eigenpairs obtained via Extended Dynamic Mode Decomposition (EDMD) as an operator perturbation of the original dynamical system. The study rigorously proves that this approximate solution corresponds exactly to a pseudo-trajectory shadowed by a true system trajectory. By establishing a precise connection among residuals, operator perturbations, and system trajectories, the paper constructs a backward stability framework for assessing Koopman eigenpairs, thereby providing a novel theoretical foundation for the credibility of data-driven methods in noisy environments.

Existing neural approaches to identifying linear dynamical systems often struggle to simultaneously ensure physical consistency and interpretability, frequently yielding unstable or unphysical solutions. This work proposes Lie Generator Networks (LGN), which enforce intrinsic stability and dissipativity by parameterizing the generator matrix in the form \( A = S - D \), where \( S \) is skew-symmetric and \( D \) is positive diagonal. By leveraging the matrix exponential for exact trajectory computation, LGN circumvents numerical integration errors. Integrating Lie algebraic structure, spectral analysis, and structured neural networks, the method accurately recovers all eigenvalues in a 100-dimensional RLC ladder circuit, achieving mean errors over two orders of magnitude lower than unconstrained baselines. Furthermore, LGN enables direct extraction of key physical quantities such as poles, natural frequencies, and damping ratios.

Traditional stability and sensitivity analyses rely on known governing equations and linearization assumptions, rendering them inadequate for nonlinear or model-unknown complex systems. This work proposes a purely data-driven framework that leverages a neural network-based dynamic simulator combined with automatic differentiation to directly extract the system’s Jacob日晚间 matrix from observational data, thereby computing eigenmodes and resolvent modes without any prior knowledge of the governing equations. The method enables, for the first time, fully automated identification of stability properties and optimal forcing responses in nonlinear systems, transcending the limitations of classical linear theory. Experiments on chaotic systems and high-dimensional fluid flows demonstrate that the framework accurately captures dominant instability modes and input–output structures even in strongly nonlinear regimes.

The Koopman operator provides a data-driven linearization framework for nonlinear dynamical systems, but its infinite-dimensionality impedes spectral estimation convergence and undermines reliability in analyzing continuous spectra and systems lacking spectral gaps. Method: We propose a unified residual error control framework, delivering the first elementary convergence proof for generalized Laplace analysis. We develop data-driven filtering power iteration, continuous spectrum identification, and spectral measure computation methods. Contribution/Results: These advances significantly enhance resolution of continuous spectra and weakly decaying modes. The resulting methodology combines theoretical rigor with numerical stability, enabling verifiable long-term forecasting and spectral decomposition. We establish a structured, pedagogically accessible standard workflow for Koopman spectral analysis—applicable to both novices and experts—that advances nonlinear system modeling from empirical fitting toward interpretable, convergent quantitative analysis.

研究通过构建Lyapunov解算子并使用Fourier神经算子来近似解决非线性系统稳定性分析中Lyapunov函数难以构建的问题。

This study addresses the long-term instability and obscure error mechanisms in neural autoregressive modeling of chaotic systems by constructing a feature analysis framework to uncover the dynamical origins of error growth. Through Jacobian spectral analysis and high-order numerical integrators, we establish an a priori stability diagnostic theory that reveals a universal linear error scaling law for integration-constrained models and proposes a stability regularization loss function. Extensive validation across 29 architectures demonstrates that this approach significantly enhances prediction accuracy and dynamical robustness. Consequently, this work provides both theoretical underpinnings and an effective optimization paradigm for neural network-based modeling of chaotic dynamics, bridging the gap between numerical stability theory and deep learning applications in complex system simulation.

This work addresses the limitation of conventional dynamical system operator estimation methods, which treat each system in isolation and fail to exploit shared dynamical structures. The authors propose DOODL, a novel framework that integrates dictionary learning with optimal transport for the first time, under the assumption that related systems approximately lie on a low-dimensional manifold in the spectral operator space. By learning a shared basis of spectral dynamical atoms, DOODL geometrically models this manifold to yield compact, interpretable system embeddings and effectively regularizes operator estimation from short or partial observations. Evaluated on Langevin dynamics and turbulent plasma simulations, DOODL reduces estimation errors by one to two orders of magnitude compared to traditional approaches under low-data regimes, substantially improving the recovery accuracy of dominant spectral dynamical structures.

本文提出了一种从数据中学习控制系统的线性算子的结构化方法，利用(半)群框架和反问题框架分析算法，以获得收敛估计器。

Accurately modeling high-dimensional nonlinear coupled systems under data-scarce conditions remains challenging. To address this, this work proposes an enhanced Extended Dynamic Mode Decomposition (EDMD) method that integrates prior knowledge of subsystem control differential equations into the learning of the Koopman operator. By explicitly embedding the governing control equations into the Koopman operator identification process—a first in the literature—the approach significantly improves modeling stability and predictive accuracy in small-sample regimes. Numerical experiments demonstrate that, compared to conventional EDMD, the proposed method achieves superior reconstruction performance and enhanced robustness on coupled oscillator systems.