matrix calculus

This study addresses the high computational cost and low efficiency of traditional methods for parameter estimation in differential equation models. To overcome these limitations, the authors propose a penalized likelihood framework based on the generalized profiling (parameter cascading) approach, which directly embeds ordinary differential equations (ODEs) into the objective function, thereby avoiding repeated numerical integration. This strategy preserves dynamic consistency while substantially improving estimation efficiency and numerical stability. As a key contribution, the work provides an open-source, reproducible Jupyter Notebook tutorial with complete code implementations covering multiple ODE modeling examples, effectively lowering the barrier to adopting advanced parameter estimation techniques and facilitating their broader application in both research and education.

This study investigates the stationary distribution and stability of stochastic differential equation systems with multimodal uncertain parameters exhibiting superposition effects, using the nonlinear Rosenzweig–MacArthur predator–prey model as a case study. For the first time, multimodal mixture-distributed parameters are incorporated into the stationary analysis of stochastic dynamical systems. System stability is quantified through the eigenvalue distribution of the Jacobian matrix, and posterior stationary density estimates are obtained via the Monte Carlo method proposed by Hoegele (2026). The results reveal that under multimodal parameter uncertainty, the system exhibits a multimodal stationary distribution, accurately delineating regions of stability. This demonstrates the effectiveness and novelty of the proposed framework for uncertainty quantification in complex ecological dynamics.

This work addresses the lack of a unified differentiable framework for inference and estimation in linear Gaussian Bayesian networks, which traditionally rely on ad hoc derivations tailored to specific graph structures. The authors propose a novel, unified differentiable covariance calculus centered on the joint covariance matrix, applicable to arbitrary directed acyclic graphs and structured parameterizations. Key tasks—including conditioning, d-separation testing, and maximum likelihood estimation with latent variables—are reformulated as linear algebraic operations on covariances, enabling gradient computation via a single backward pass. By integrating K-recursive covariance mappings, automatic differentiation, and information geometry, the framework naturally subsumes the Slepian–Bangs formula and the Cramér–Rao bound. Its correctness and consistency are validated on state-space models and their extensions with skip connections.

This work addresses the fragmentation in existing Gaussian process–based approaches for approximating solutions to differential equations by proposing a unified Bayesian probabilistic framework. The framework embeds differential equation constraints into the likelihood function through derivative matching, thereby enabling simultaneous estimation of unknown parameters and quantification of uncertainty in the solution. It systematically integrates several established Gaussian process methods for the first time, elucidating their underlying connections. The generality and efficacy of the proposed approach are demonstrated across multiple benchmark problems, establishing a coherent foundation for future theoretical advancements and practical applications in this domain.

Existing mathematical modeling lacks a rigorous, unambiguous ontological foundation, hindering a unified characterization of the mapping between models and real-world phenomena. This paper introduces, for the first time, an axiomatic definition of mathematical models grounded in Hilbert-space operator theory: a model is formalized as a computable operator acting on random variables, systematically unifying theoretical derivation, experimental implementation, and statistical identification. We further establish a geometric correspondence between the model manifold and the prediction surface, exposing intrinsic structural properties and the fundamental nature of model computability. This framework fills a critical gap in the formal ontology of modeling, providing a unified mathematical foundation for interdisciplinary model construction. It significantly enhances the logical rigor of theoretical inference and the reliability of empirical validation.

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.

This work addresses the challenge of jointly inferring multi-level functional states—such as curves, derivatives, and integrals—in functional data modeling, where existing methods struggle to account for derivative uncertainty, cross-level covariance, and identifiability of integration constants. The authors propose an anchored Gaussian process differential ensemble framework that explicitly models integration constants by embedding anchor points together with their mean-square derivatives and repeated integrals into a joint Gaussian state, enabling efficient computation via transformed Hilbert spaces. A key innovation is the separation of anchor-induced covariance from boundary uncertainty, revealing that integration constants cannot be uniquely identified from anchors alone. To enhance derivative recovery accuracy, the method introduces the TARTARE calibration strategy. Theoretical analysis combines Laplace–Dirichlet basis functions, finite-rank approximations, and operator-level approximation bounds. Experiments demonstrate substantially improved posterior derivative estimation in second-order simulations while preserving accuracy in anchors and integrals, and a motorcycle crash case study confirms coherent inference of coupled kinematic states and functional turning points.

Bayesian parameter inference for ordinary differential equation (ODE) models from observational data often neglects discretization error introduced by numerical solvers, leading to overconfident and biased posterior estimates. Method: We propose a joint uncertainty quantification framework that models discretization error as a time-evolving stochastic process. Leveraging an asymptotically justified Markov prior, we explicitly encode its variance structure; integrating a state-space model with randomized numerical solvers enables simultaneous Bayesian inference of both ODE parameters and discretization error. Contribution/Results: Experiments demonstrate that our approach significantly broadens the support of the parameter posterior distribution while accurately disentangling and quantifying uncertainties attributable to parameters versus discretization. This enhances robustness and interpretability of ODE models under sparse and noisy observations.

Existing approaches are largely confined to Euclidean spaces and struggle to perform statistical inference on differential quantities—such as directional derivatives and curvature—for spatial processes defined on Riemannian manifolds. This work proposes the first differential inference framework for Gaussian processes on compact Riemannian manifolds, jointly modeling the original process along with its derivative and curvature processes to formalize smoothness. It establishes necessary conditions on kernel functions for the existence of derivatives and develops a theory of validity for multivariate processes. Leveraging Riemannian geometry, differential operators on vector fields, and kernel methods, the framework enables predictive inference from partial observations. Simulations on polyhedral meshes demonstrate accurate derivative estimation, thereby opening new avenues for Gaussian process modeling on non-Euclidean manifolds.

This work proposes a unified mathematical framework grounded in dynamic information flow for constructing structurally rigorous models of future prediction. By integrating filtering theory, regular conditional probabilities, Markov semigroups, infinitesimal generators, and multiple information geometries—including Hilbert, Fisher–Rao, and Wasserstein—the approach conceptualizes prediction as the construction of conditional distributions governed by informational, geometric, and modeling constraints. The framework elucidates deep connections among classical results such as the tower property and semigroup laws, as well as Itô’s formula and backward equations. Explicit transition laws, spectral decompositions, term structures, and asymptotic behaviors are derived within canonical models like Ornstein–Uhlenbeck and Cox–Ingersoll–Ross, thereby establishing a compact mathematical mapping from idealized theoretical constructs to empirical forecasting.

This work proposes a unified variational generative modeling framework based on stochastic differential equations (SDEs) to efficiently address complex data generation tasks, including images, videos, and biomolecular structures. By incorporating both ordinary and stochastic differential equations, the authors derive the evidence lower bound (ELBO) from a variational inference perspective, systematically demonstrating that diffusion models, score matching, and flow matching are distinct parameterizations within this general framework. Through theoretical analysis grounded in the Fokker–Planck equation and empirical validation via one-dimensional density modeling experiments, the study provides clear comparisons among different parameterization strategies, confirming the proposed framework’s theoretical coherence, expressive capacity, and practical efficacy.