mean-field dynamical reduction

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.

Reduced-order modeling of spatiotemporal chaotic systems faces dual challenges—physics model mismatch and scarce training data. Method: We propose a physics-informed, data-driven hybrid framework: (i) an autoencoder learns a low-dimensional invariant manifold; (ii) the full-order model’s vector field is orthogonally projected onto this manifold; (iii) a differentiable neural ordinary differential equation (Neural ODE) predictor is constructed on the manifold and augmented with a Bayesian error correction mechanism for uncertainty quantification and online adaptation. Results: Experiments on the Kuramoto–Sivashinsky and complex Ginzburg–Landau equations demonstrate that our method significantly outperforms purely data-driven approaches, maintaining high-fidelity long-term predictions under adverse conditions—including parametric mismatch and sparse observations. This work establishes a new paradigm for robust physics-informed deep learning in chaotic system modeling.

This study addresses the challenge of reduced-order modeling for non-autonomous nonlinear dynamical systems by proposing a latent-space Lagrangian framework. The method jointly learns latent coordinates and an energy network, incorporating the principle of virtual work to ensure physical consistency. Furthermore, a force supervision mechanism replaces conventional ODE solvers, thereby eliminating the reliance on numerical integration during training. Experimental results demonstrate that the proposed framework accurately captures complex nonlinear dynamics while exhibiting strong generalization to unseen external forces and initial conditions.

This study addresses the fundamental challenge in multiscale dynamical systems modeling: *how to automatically identify a small set of critical slow variables to construct interpretable and predictive low-dimensional models*. To this end, we propose a data-driven dimensionality reduction framework grounded in the information bottleneck principle. Methodologically, we establish, for the first time, an analytical connection between slow variables and eigenfunctions of the Koopman (or transfer) operator; introduce an optimal truncation criterion based on information compression rate; and integrate variational inference with autoencoding neural networks to build an interpretable deep learning architecture capable of discovering emergent order parameters. Applied to satellite atmospheric flow videos, our method successfully extracts dominant slow variables; applied to experimental videos of cyanobacterial microcolonies, it uncovers a novel synchronization order parameter. The framework thus achieves a unified balance between model interpretability and predictive accuracy.

This work addresses the limited understanding of how disorder influences training and generalization in high-dimensional dynamical systems relevant to machine learning. By integrating dynamical mean-field theory (DMFT), random matrix theory, the cavity method, and path integrals, the authors reduce complex high-dimensional coupled systems to effective single-site stochastic processes driven by non-Hermitian random matrices. They uncover a novel mechanism—rooted in non-Hermitian structure—that leads to non-monotonic training loss dynamics in settings such as gradient flow, random feature models, and deep linear networks. A DMFT-based bias–variance decomposition is introduced via ensemble averaging over noise realizations, and the emergent spiked random matrix structure underlying feature learning in deep linear networks is characterized. Finally, asymptotic dynamics of both training and test losses are derived for high-dimensional random data, providing a theoretical foundation for quantifying strategies like ensemble learning.

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 computational expense of high-fidelity simulations in parametric stochastic dynamical systems and the inability of existing reduced-order models to adequately handle stochasticity or quantify uncertainty. The authors propose a data-driven framework that jointly learns a probabilistic autoencoder and a stochastic differential equation in the latent space via amortized stochastic variational inference, yielding a continuous-time stochastic reduced-order model capable of generalizing to unseen parameters and forcing conditions. By introducing a Markovian Gaussian process reparameterization, the method eliminates the need for costly forward solvers, enables incorporation of physical priors, and achieves computational complexity independent of both data size and system stiffness. Evaluated on three challenging benchmarks, the approach significantly outperforms state-of-the-art methods, demonstrating breakthroughs in both generalization capability and computational efficiency.

This work addresses the challenge of accurately identifying governing equations of complex dynamical systems under extreme data scarcity and high acquisition costs. The authors propose a novel active learning strategy that integrates active sampling with sparse dynamics discovery for the first time. By leveraging ensemble SINDy (E-SINDy) to quantify epistemic uncertainty, the method iteratively selects and acquires new measurements in the most informative regions of the state space. This approach substantially enhances the efficiency of discovering governing ordinary and partial differential equations under severely limited data budgets. Validation on benchmark systems—including Lorenz, Burgers’, and Kuramoto–Sivashinsky—demonstrates that the method accurately recovers the true equations using significantly fewer data points than random sampling, even across varying noise levels.

This work addresses the failure of traditional mean-field approximations in modeling social behavior dynamics under high-noise networked settings by proposing a method based on the weak-form sparse identification of nonlinear dynamics (WSINDy). The approach directly learns continuous-time ordinary differential equation models from noisy trajectory data collected under multiple initial conditions, capturing the evolution of online-offline coupled social behaviors without relying on mean-field assumptions. By leveraging the weak formulation, the method recovers high-fidelity, interpretable dynamical models directly from stochastic process data. Experimental results demonstrate that incorporating only a few additional initial conditions substantially improves modeling accuracy in high-noise environments, significantly outperforming conventional mean-field approaches.

This work addresses the lack of verifiability in learned world models when deployed in high-assurance systems by proposing a novel framework that integrates classical model order reduction (MOR) with modern world modeling. The approach combines proper orthogonal decomposition (POD) with an encoder–decoder architecture, incorporates physics-informed error bounds derived from physical priors, and employs measurement-driven action-conditioned modeling to ensure verifiable closed-loop predictions, exceptional data efficiency, and physical consistency. By systematically unifying MOR theory with contemporary world model paradigms, this study establishes a new modeling methodology that simultaneously achieves reliability and performance for safety-critical applications.