analytical integration

This work addresses parametric numerical integration problems—including statistical functional evaluation, Chebyshev spectral approximation, and integrals arising from differential equations—by proposing a derivative-supervised differentiable machine learning framework. The method explicitly incorporates analytical derivative information into surrogate modeling of integrals for the first time, overcoming the limitations of conventional black-box regression while preserving physical consistency and substantially improving accuracy and generalization. Technically, it integrates deep neural networks with derivative-augmented training to enable efficient inference in high-dimensional parameter spaces. Evaluated on diverse benchmarks encompassing both smooth and ill-conditioned integrals, the approach achieves over 40% average reduction in mean squared error compared to standard architectures, while reducing sample requirements by a factor of 3–5. It thus delivers high accuracy, strong scalability, and exceptional sample efficiency.

This study addresses accuracy and consistency issues in the numerical construction of the Karhunen–Loève expansion (KLE) arising from discretization, quadrature rules, and finite sample sizes. It establishes an algebraic equivalence between the spectral decomposition of the Fredholm integral equation and the singular value decomposition (SVD) of a weighted sample covariance matrix, thereby unifying model-driven and data-driven KLE frameworks. The work innovatively constructs the covariance function on a non-simply-connected three-dimensional toroidal domain using the shortest interior path distance, and implements the approach numerically with unstructured meshes and Gaussian quadrature. Experiments demonstrate that, in a one-dimensional benchmark problem, SVD-based eigenvalue estimates and empirical KL coefficients converge to the theoretical 𝒩(0,1) distribution. In two-dimensional irregular and three-dimensional toroidal domains, the study systematically quantifies the combined influence of discretization strategy, quadrature accuracy, and sample size on KLE reconstruction error.

This paper addresses the lack of reliable upper and lower bounds for moment integrals—such as variance estimation in Bayesian inference—under scalar, unnormalized probability distributions. We propose a sequential bounding method grounded in the majorization-minimization (MM) framework and envelope principles. Our approach iteratively refines bounds to yield strictly monotonic and provably convergent sequences of upper and lower bounds. This work is the first to systematically apply the MM paradigm to order-preserving bound construction for moment estimation, ensuring both theoretical convergence guarantees and user-controllable accuracy. Unlike conventional numerical integration or Monte Carlo methods, our technique rigorously preserves inequality ordering, thereby enhancing reliability with formal theoretical assurances in signal processing and Bayesian uncertainty quantification. Experiments demonstrate significant improvements over existing bounding strategies in variance estimation tasks.

This paper addresses the challenge of analytically characterizing the conditional and unconditional density functions for affine jump-diffusion models with state-independent jump intensities. We propose a novel density approximation method based on closed-form moment recursion and generalized moment matching. For the first time, we derive explicit recursive formulas for all orders of conditional and unconditional moments of such processes and construct a density approximation that admits an analytical expression—up to the normalization constant. Compared to conventional Monte Carlo simulation, our method achieves comparable accuracy in option pricing and path simulation while improving computational efficiency by an order of magnitude. This advancement significantly extends the practical applicability of affine jump-diffusion models in financial derivative pricing and high-efficiency stochastic simulation.

Bayesian posteriors are often skewed, whereas mainstream deterministic approximations—such as Laplace’s method and variational Bayes—rely on symmetric densities (e.g., Gaussians), leading to systematic bias and reduced accuracy. Method: We propose a generic, optimization-free skewness-aware perturbation framework that can be seamlessly integrated with any off-the-shelf symmetric approximation. Our approach constructs analytical perturbations based on skew-symmetric density families, unifying asymptotic expansion and variational analysis. Contribution/Results: We theoretically establish finite-sample accuracy improvement and prove that the asymptotic convergence rate is accelerated by at least a factor of √n. The method is model-agnostic and compatible with diverse symmetric approximation paradigms. Numerical experiments demonstrate substantial gains over standard Gaussian approximations—particularly in moderate-to-small sample regimes and under strong posterior skewness—empirically validating the predicted convergence acceleration and robustness.

This work addresses inverse problems in science and engineering—such as parameter inference and detector response unfolding—by proposing a unified simulation-based inference (SBI) framework that systematically integrates Bayesian and frequentist perspectives. Leveraging machine learning techniques, including neural posterior estimation and neural likelihood estimation, the framework enables efficient and general-purpose parameter inference, with extensions to empirical Bayes and unfolding tasks. The paper provides a comprehensive review of SBI methodologies and their application paradigms, while also offering a thorough analysis of validation strategies and inherent limitations. By clarifying best practices and pitfalls, this study advances the reliable deployment and innovative application of SBI in scientific domains.

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.

This work addresses the efficient computation of exponential expectation integrals of the form $ \mathbb{E} \exp\left\{\sum_{i=1}^m \phi_i\right\} $ under standard Gaussian or symmetric exponential measures in high-dimensional spaces, where each function $ \phi_i $ depends only on a small subset of coordinates. By leveraging the local dependency structure of the functions together with their Lipschitz constants, the study establishes, for the first time, verifiable sufficient conditions ensuring that the integral remains bounded away from zero. The approach integrates probabilistic measure analysis, Lipschitz estimates, and dependency graph modeling to provide a novel theoretical foundation for the non-degeneracy of high-dimensional integrals. This framework is successfully applied to problems in high-dimensional volume estimation and lattice point counting in polytopes, significantly enhancing computational feasibility.

This work addresses the challenge of verifying mathematical proofs generated by large language models by formally encoding, for the first time, an entire advanced undergraduate probability textbook—including its measure-theoretic foundations—into Lean. To bridge the semantic gap between the textbook’s exposition and the abstract formalism of the Mathlib library, the authors introduce an “interface lemma” strategy. Combined with structured proof engineering and formalization techniques specific to measure theory, this approach yields a reusable, machine-verifiable infrastructure spanning fourteen textbook chapters. The resulting formalization not only provides rigorous verification of all stated theorems and explicit articulation of their assumptions but also establishes a robust foundation for reliable AI-assisted mathematics, educational applications, and future formalization efforts in probability theory.

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.