linear response theory

This study addresses the challenge of reliably identifying causal relationship chains from observational data in stochastic nonlinear systems. We propose a novel framework integrating physical response theory with modern machine learning: leveraging linear response theory as a physics-informed constraint, and combining statistical learning with stochastic process analysis to construct an interpretable and generalizable causal discovery method. We establish, for the first time, the asymptotic efficiency of linear response-based causal predictors in large-scale Markov networks and derive their performance bounds. The method ensures theoretical rigor while maintaining data-driven adaptability. It successfully reconstructs causal structures and enables accurate prediction across diverse linear and nonlinear stochastic systems—including high-dimensional and non-Gaussian settings—demonstrating substantial improvements in both accuracy and scalability of causal inference for complex dynamical systems.

How do complex nonlinear network systems exhibit emergent macroscopic linear behavior arising from microscopic nonlinear dynamics? Method: We propose and rigorously prove that the spatially averaged dynamics of large-scale heterogeneous networks asymptotically linearize when inter-subsystem spatial correlations decay with distance. Our analysis integrates mixed-sequence theory, asymptotic analysis, finite-sample convergence rate estimation, and spatial embedding modeling to systematically characterize both the conditions for linearization and the resulting dynamical properties. Contribution: We establish the first general theoretical framework for macroscopic linear emergence, providing rigorous sufficient conditions for linearization. The framework encompasses linear time-invariant limiting dynamics, explicit finite-sample convergence rate bounds, and broad applicability to diverse real-world networks—including neural, ecological, and multi-agent systems—thereby overcoming classical limitations of local approximations and homogeneity assumptions in conventional linearization approaches.

This work exposes a fundamental limitation of conventional sparse optimization-based equation discovery for modeling chaotic systems: equations inferred from different measurements—though capable of generating highly similar chaotic attractors—lack uniqueness and physical interpretability, leading to potentially misleading inferences. Integrating sparse regression, Koopman spectral analysis, and numerical simulations, the study systematically examines multiple chaotic systems and reveals that small-magnitude Koopman eigenvalues are highly sensitive to measurement perturbations, whereas only large-magnitude eigenvalues remain robust. This undermines the prevailing “unique correct equation” paradigm. The key contribution is the first operator-spectral proof that the deterministic assumption underlying equation discovery fails for chaotic dynamics. Consequently, the paper argues that for strongly nonlinear, highly sensitive systems, end-to-end data-driven modeling—particularly machine learning approaches—should supersede the pursuit of explicit differential equations.

This work addresses the problem of unsupervised learning of low-dimensional, manipulable dynamical system representations—namely, compact and smooth state variables coupled with differentiable vector fields—directly from raw video, without prior physical knowledge or domain-specific assumptions. We propose the first end-to-end, video-driven framework for manipulable dynamics discovery, integrating neural implicit state modeling, contrastive spatiotemporal regularization, and differential-geometric constraints to jointly ensure state interpretability, dynamical differentiability, and behavioral analyzability. Evaluated across diverse dynamical systems—including chaotic, limit-cycle, stable fixed-point, and natural oscillatory regimes—the method accurately recovers essential dynamical features (e.g., attractors, bifurcations, conserved quantities) and achieves significantly higher long-horizon prediction accuracy than existing baselines.

This work addresses the problem of jointly identifying dynamical laws and causal structure from observed time-series data generated by ordinary differential equation (ODE)-driven systems, while enabling counterfactual prediction under interventions. We propose a novel neural ODE framework that—uniquely—integrates lightweight sparsity and symmetry regularization to achieve robust dynamics modeling and causal graph learning even under non-identifiable conditions. The method unifies the representation of inter-variable dynamics and causal dependencies, supporting explicit intervention inference on both variables and system parameters. Evaluated across diverse synthetic benchmarks—including linear and nonlinear first- and second-order ODE systems—as well as real-world datasets, our approach significantly improves dynamical reconstruction accuracy and counterfactual prediction reliability, while enhancing causal interpretability through structured, sparse, and symmetric Jacobian estimation.

This study addresses the need for a unified understanding of the relationships and generalization capabilities across data-driven modeling paradigms, ranging from classical inverse problems to modern neural operators. By integrating inverse problem theory, sparse identification of dynamical systems, neural ordinary differential equations, and neural operators—and further incorporating the philosophical notion of “mechanism” from philosophy of science—the authors construct a cohesive analytical framework. The work demonstrates that genuine mechanistic discovery and robust cross-scenario generalization are achievable only when models recover concise differential equation structures underlying the observed data. This perspective clarifies the fundamental connections among diverse modeling approaches and underscores the critical role of mechanistic interpretability in enabling reliable generalization, thereby offering a theoretical foundation for the categorization, selection, and design of scientific machine learning models.

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.

This study investigates the trade-offs among collective output, stability, and adaptability in multi-agent systems to achieve optimal order. To this end, it proposes a unified analytical framework grounded in agent influence (power) and response functions, incorporating task dependency and system relativity into formal measures of order, entropy, and information. The framework reveals an intrinsic trade-off between synchrony and system fragility. By integrating multi-agent modeling, macroscopic variable derivation, and risk-preference parameterization, the work optimizes system utility, thereby enhancing the predictability and controllability of collective behavior. It further delineates the precise conditions under which collective intelligence emerges and optimal order is attained.

This work addresses the lack of a unified computational framework for analyzing non-ergodicity, modeling heavy-tailed dynamics, and studying decision-making under uncertainty in stochastic processes. To this end, we introduce an open-source Python library that, for the first time, integrates non-ergodicity diagnostics, simulation of heavy-tailed processes—such as multiplicative Lévy growth and memory-dependent mean-reverting dynamics—and agent-based experimentation within a single platform. Built upon the scientific Python ecosystem (NumPy/SciPy), the library supports end-to-end workflows including stochastic process definition, simulation, parameter inference, and partial solution of stochastic differential equations. Through several reproducible examples—ranging from heavy-tailed ensemble diffusion to pre-asymptotic fluctuation analysis—it substantially reduces boilerplate code and enhances both reproducibility and development efficiency in the study of time-averaged behaviors of complex stochastic systems.

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.