fast-slow ode analysis

This work addresses two key challenges in multiscale dynamical systems: the difficulty of initializing slow manifolds and the high computational cost of computing steady-state solutions for bifurcation diagrams. We propose a geometry-driven inverse modeling framework based on conditional score-based generative models (cSGMs). For the first time, conditional generative modeling is applied to slow manifold sampling and bifurcation diagram interpolation—enabling high-fidelity steady-state initial conditions to be generated directly from prescribed slow-variable values or new parameter configurations, without explicitly solving differential equations. The method integrates manifold learning, dynamical system dimensionality reduction, and generative inverse modeling to achieve label-controllable, mesh-free, data-driven slow manifold initialization and bifurcation diagram extrapolation and completion. Extensive validation across multiple ODE and PDE systems demonstrates substantial acceleration in steady-state acquisition while preserving accuracy, generalizability, and computational efficiency.

Modeling incompressible flows governed by the Navier–Stokes equations near Hopf bifurcation points remains challenging due to high computational cost and sensitivity to parameter variations. Method: This paper proposes a simulation-free parametric reduced-order model (ROM) by directly applying invariant manifold parametrization (Parameterized-Continuation, PC) theory to the Navier–Stokes equations. Leveraging analytical nonlinear dynamical system reduction, it constructs a low-dimensional manifold intrinsically embedding the original dynamics—without requiring full-order simulations or training data. Contribution/Results: The method enables single-point computation for multi-parameter generalization, accurately reproducing both pre-bifurcation steady states and post-bifurcation limit-cycle behavior, with results closely matching full-order simulations. Computational cost is drastically reduced. By eliminating reliance on data-driven learning or repeated high-fidelity simulations, this work establishes a new paradigm for real-time analysis and control of Hopf-bifurcating flows.

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 data-driven modeling of physical systems—including autonomous and forced dynamical systems—by introducing a novel dynamics learning framework grounded in invariant foliation theory. Methodologically, it extends invariant foliation theory for the first time to parameter-dependent and volume-preserving forced systems; proposes a multi-neighborhood foliation fusion strategy to robustly reconstruct invariant manifolds; and integrates normal-form transformations to enable interpretable extraction of instantaneous frequency and damping characteristics. The contributions are threefold: (i) it effectively mitigates overfitting and underfitting, achieving high-fidelity reconstruction of invariant manifolds from sparse single- or multi-trajectory data; (ii) it delivers accurate long-term predictions across autonomous, periodically forced, quasiperiodically forced, and chaotic forced systems; and (iii) it offers strong interpretability and quantitative physical insight, establishing a new paradigm for low-data, high-accuracy, and physically interpretable dynamical modeling.

This work addresses the learning and reduced-order modeling of invariant manifolds (IMs) in discrete dynamical systems. Methodologically, it proposes a physics-informed hybrid neural-analytic approach that couples shallow neural networks with orthogonal polynomial (Legendre/Chebyshev) power series. This integration synergistically combines the local analytic guarantees of polynomials with the global representational capacity of neural networks—ensuring strict exponential convergence near fixed points while overcoming the limited radius of convergence inherent to purely polynomial methods. Innovatively, it is the first framework to unify analyticity constraints, numerical stability optimization, and physics-informed embedding within IM learning. Evaluated on three benchmark problems, the method achieves significantly higher approximation accuracy than pure polynomial or pure neural network baselines, maintains controllable training cost, and exhibits markedly improved convergence robustness.

This work addresses the challenge of learning stochastic multiscale systems with unobserved fast processes from a single trajectory of slow variables, where the invariant distribution of the fast dynamics is unknown. The authors propose an end-to-end learning framework based on stochastic differential equations that integrates stochastic averaging for structure-preserving dimensionality reduction. Crucially, they introduce normalizing flows to flexibly parameterize the invariant distribution of the latent fast variables—a first in this context—and combine this with variational Bayesian inference to quantify epistemic uncertainty in model parameters. Using only a single observed slow-variable trajectory, the method accurately identifies the effective stochastic dynamics, achieving both scalability and significantly enhanced modeling robustness and reliability of uncertainty quantification.

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 high computational cost and poor robustness commonly encountered in spectral submanifold (SSM)-based reduced-order modeling of high-dimensional nonlinear systems. It reveals for the first time that SSMs inherently possess equivariance properties and leverages this insight to propose an equivariant spectral submanifold (eSSM) framework. By explicitly embedding the symmetries of the full-order model, eSSM integrates physical priors with group-action structures to construct an efficient nonlinear dimensionality reduction approach. Grounded in group representation theory, the method introduces equivariant manifold modeling and an associated SSM reduction algorithm. Numerical experiments across multiple benchmark problems—including tests within the Scientific Machine Learning Universal Tasks framework—demonstrate significant improvements in computational efficiency, accuracy, and numerical stability compared to conventional approaches.

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 limitation of conventional Hamiltonian Neural Networks (HNNs) in modeling multi-timescale dynamical systems due to spectral bias. The authors propose Frequency-Separable Hamiltonian Neural Networks (FS-HNN), which explicitly decompose the Hamiltonian into fast and slow dynamical components, each modeled by dedicated subnetworks. These subnetworks are jointly trained on multiscale sampled data, enabling reconstruction of the full Hamiltonian structure without requiring strong domain-specific assumptions. The framework is further extended to partial differential equations by learning state- and boundary-condition-dependent symplectic operators. Evaluated across diverse ordinary and partial differential equation systems, FS-HNN demonstrates significantly improved long-term extrapolation accuracy and superior generalization capability across multiple timescales.