perturbation bounds

Current theory lacks a quantitative characterization of the conditions and sample complexity required for uniquely and stably identifying the true ordinary differential equation (ODE) from observed solution data. This work proposes the Hausdorff distance between solution sets as a natural metric for comparing ODEs and establishes, for the first time, an identifiability framework based on this metric, encompassing both linear and nonlinear systems with Lipschitz or Hölder continuous vector fields. By leveraging metric entropy estimates, minimax analysis, and function space theory, the study derives precise learnability conditions, sharp distinguishability criteria, and upper bounds on sample complexity for classes of ODEs. These results provide rigorous theoretical guarantees for equation discovery tasks and fill a critical gap in the theoretical understanding of ODE identifiability within scientific machine learning.

This work addresses the challenge of evaluating generalization performance in quantized dynamical system identification, where data dependence and non-ideal optimization complicate theoretical analysis. The authors propose a unified framework for statistical error bounds, leveraging a block decomposition technique to derive slow-rate bounds and introducing a novel subsampled interval strategy to establish fast-rate, variance-adaptive bounds. These bounds explicitly link the number of bits used for model quantization to statistical complexity. Notably, this is the first study to incorporate hardware constraints—such as quantization bitwidth—into generalization error theory, offering theoretically grounded, interpretable, and practically actionable guarantees for real-world applications including quantized modeling and hybrid system identification.

This work addresses the structural identifiability of parameters in stochastic differential equation (SDE) models under multiple interventions—i.e., whether SDE parameters can be uniquely recovered from samples of post-intervention stationary distributions. Theoretically, we establish the first uniqueness guarantee for SDE parameter recovery under multi-intervention settings; for linear SDEs, we derive a tight lower bound on the minimum number of required interventions; for weak-noise nonlinear SDEs, we obtain an upper bound on identifiability. Methodologically, we propose a parametric framework featuring learnable activation functions, integrating intervention modeling, stationary distribution analysis, and weak-noise asymptotic theory. Experiments on synthetic data demonstrate that our approach accurately recovers ground-truth parameters, and the theory-guided learnable architecture significantly improves both estimation accuracy and robustness.

Existing finite-sample error bounds in linear system identification are significantly loose due to overestimation of the influence of state dimensionality. This work reveals, via the central limit theorem, that ordinary least squares (OLS) estimation incurs error bounds whose looseness scales proportionally with the state dimension. To address this, the paper introduces a novel second-order error decomposition that leverages a matrix-valued martingale structure to characterize lower-order terms. As a result, it establishes non-asymptotic error bounds that are constant-factor tight in the Frobenius norm and depend on the state dimension only through a polylogarithmic factor in the spectral norm. For stable systems and multi-trajectory settings, these bounds approach instance-optimal rates, markedly improving upon the tightness of existing results.

This work addresses the problem of estimating the drift function in stochastic differential equations when the diffusion coefficient is known, framing it as a denoising task amenable to diffusion-based modeling. By leveraging conditional score matching, the method recovers the drift function from discrete observations across multiple sample trajectories. The study establishes, for the first time, an explicit time-averaged mean squared error risk bound for this class of estimators. The theoretical analysis elucidates the interplay among four key sources of error: Euler–Maruyama discretization, score approximation, noise initialization, and sampling variance. The resulting risk decomposition provides a sharp characterization of how various hyperparameters influence estimation accuracy, thereby offering rigorous theoretical guarantees for drift estimation powered by diffusion models.

This work addresses the long-standing perception that zeroth-order (ZO) optimization algorithms inherently converge more slowly than first-order (FO) methods due to dimensionality dependence. By interpreting ZO algorithms through the lens of dynamical systems, the paper models them as perturbed averaged versions of FO algorithms and introduces, for the first time, the input-to-state stability (ISS) theoretical framework to analyze their convergence behavior. Under suitable conditions, the authors rigorously establish that ZO methods can eliminate the dimensionality gap in expectation, converging at the same rate as FO methods to an arbitrarily small neighborhood of the FO fixed point. These theoretical findings are corroborated by numerical experiments, highlighting the underappreciated convergence potential of ZO optimization algorithms.

This work addresses the structural identifiability of ordinary differential equation (ODE)-based mechanistic models—specifically, whether model parameters can be uniquely determined from ideal observational data—and proposes a unified symbolic analysis framework implemented in Julia. Built upon the StructuralIdentifiability.jl package, the framework integrates symbolic computation with parameter-output mapping analysis to support assessments of local and global identifiability, observability, and extraction of identifiable parameter combinations. As the first fully reproducible tutorial within the SciML ecosystem, it not only enables model reparameterization and informs experimental design but also demonstrates its efficacy across seven representative case studies spanning epidemiology, pharmacokinetics, and other domains, thereby offering both a practical workflow and theoretical foundation for modeling complex dynamical systems.

This work addresses the limitations of classical algorithmic stability theory, which typically relies on strong assumptions such as bounded loss functions or sub-Gaussian/sub-Weibull tail behavior—conditions often violated in heavy-tailed or unbounded loss settings. The paper introduces a novel $L_p$ stability framework that requires only finite $L_p$ moments of the loss function, thereby relaxing the conventional bounded differences condition. By extending McDiarmid’s inequality to accommodate $L_p$ constraints, the authors derive sharp high-probability generalization bounds under this significantly weaker assumption. This approach is shown to be broadly applicable across multiple learning paradigms, including empirical risk minimization, transductive regression, and meta-learning, demonstrating that robust generalization guarantees can still be achieved even when losses are unbounded, provided $L_p$ stability holds.

This work presents the first fully AI-generated, rigorous explicit lower bounds for the advection-diffusion equation across three distinct flow regimes—non-diffusive shear flows, diffusive shear flows, and rapidly oscillating time-periodic flows—without any human intervention. Leveraging the multi-agent automated reasoning system QED, which integrates partial differential equation analysis with formal verification, the study derives data-dependent explicit constants: a polynomial Ḣ⁻¹ decay bound for non-diffusive shear flows, a uniform positive lower bound on mixing scales for diffusive shear flows, and an exponential L² decay bound for rapidly oscillating flows. This achievement marks a breakthrough in the application of artificial intelligence to complex mathematical proofs.

This study addresses the problem of global parameter identifiability for linear ordinary differential equation models in which parameters depend linearly on state variables and rationally on other parameters. By reformulating identifiability as the injectivity of the corresponding input–output map and leveraging tools from algebraic geometry and differential algebra, the authors establish—for the first time—that this identifiability problem is NP-hard. This result resolves a longstanding gap in the computational complexity theory of identifiability for this important class of models, providing a rigorous lower bound on its intrinsic computational difficulty. Consequently, it offers a solid theoretical foundation for the design and performance evaluation of future algorithms aimed at tackling parameter identifiability in such systems.