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This work addresses the limitations of current automatic formalization research, which predominantly focuses on well-supported mathematical domains and relies solely on kernel acceptance rate as a quality metric, thereby neglecting the practical needs of underrepresented areas such as numerical analysis and lacking comprehensive evaluation. For the first time, we employ a Lean 4 coding agent to formalize an entire textbook—*Numerical Methods for Ordinary Differential Equations*—from scratch and introduce a three-dimensional evaluation framework that jointly assesses semantic correctness, Mathlib reusability, and cross-file reusability. Through LLM-as-judge, semantic validation, and dependency analysis, we uncover pervasive issues in existing systems, including incomplete statements and weakened assumptions, demonstrating that kernel acceptance rate substantially overestimates formalization quality. Our approach establishes a reproducible, multidimensional auditing paradigm for trustworthy automated formalization.

This work addresses parameter estimation for rational ordinary differential equation (ODE) models. Traditional homotopy continuation methods often fail on ill-conditioned or high-dimensional instances. To overcome this, we propose a novel certified algebraic approach that— for the first time—integrates Rational Univariate Representation (RUR) and real root isolation techniques into ODE parameter estimation, transforming parameter recovery into a rigorously verifiable polynomial system solving problem. Our method combines differential-algebraic elimination, rational interpolation, and RUR construction, and employs HomotopyContinuation.jl for hybrid numerical–symbolic solving. Experiments demonstrate that our RUR-based solver successfully handles multiple challenging benchmarks where homotopy continuation fails, significantly improving estimation accuracy and reliability. Moreover, the results reveal complementary strengths between RUR and homotopy methods, thereby expanding the applicability frontier of certified parameter estimation for rational ODE systems.

This work proposes a systematic framework to enhance the computational efficiency and numerical stability of evaluating high-degree matrix polynomials. Specifically, for polynomial degrees eight and higher, the method generates and validates stable coefficient sets that reduce the number of required matrix multiplications by one compared to the classical Paterson–Stockmeyer scheme. To address instability issues in the original formulation, the authors introduce structural variants and design a reliability metric to assess the expected numerical accuracy of candidate coefficient sets. Nonlinear polynomial systems are solved using variable-precision arithmetic (VPA), and an in-house tool, MatrixPolEval1, enables efficient screening and validation. Applied to matrix exponentials and geometric series, the approach achieves a saving of one matrix multiplication while maintaining comparable numerical accuracy.

This work addresses the challenge of parameter estimation in physical system simulation models arising from complex residual distributions. The authors propose an end-to-end estimation framework that employs embedded normalizing flows to map intricate residuals onto a simple base distribution. Indirect constraints are imposed on this base distribution through empirical likelihood under moment conditions. Model and flow parameters are jointly optimized via implicit differentiation combined with first-order gradient methods. By innovatively integrating normalizing flows with constrained empirical likelihood, the approach establishes an information-theoretically interpretable and computationally tractable framework. The resulting inverse transformation serves as an invertible surrogate model, enhancing both the accuracy and efficiency of parameter estimation while enabling quantification of model bias and sensitivity analysis.

This work addresses the convergence guarantees of stochastic line search optimization for over-parameterized models under interpolation conditions. We establish a necessary and sufficient condition on the search direction—applicable to a broad class of methods—that ensures finite termination and bounded backtracking steps, and rigorously prove linear convergence under the Polyak–Łojasiewicz (PL) assumption. The condition unifies major first-order strategies—including momentum, conjugate gradient, and adaptive preconditioning—providing a verifiable theoretical foundation for their principled integration with stochastic line search. Our analysis fills a critical gap in the convergence theory of stochastic line search methods and significantly extends both the applicability and reliability of efficient first-order optimization in interpolation learning regimes.

This study addresses the high sensitivity to noise inherent in differential-algebraic parameter estimation, which arises from its reliance on exact derivatives and severely limits practical applicability. To overcome this limitation, this work integrates Gaussian process regression (GPR) into the differential-algebraic framework, proposing a robust parameter estimation method for ordinary differential equations that synergizes GPR with algebraic elimination. Furthermore, a first-order error analysis theoretical framework is established to characterize noise propagation and parameter sensitivity. Benchmark evaluations demonstrate that the proposed approach achieves state-of-the-art performance, successfully recovering parameters within a 10% relative error in 88.5% of experimental runs. These results confirm that the method effectively resolves the challenge of parameter identification from noisy observational data.

This work addresses the challenge of selecting the regularization parameter (nugget) in ill-posed linear systems arising in machine learning, where existing adaptive methods lack compatibility with automatic differentiation and suffer from computational inefficiency. To overcome these limitations, we introduce autonugget, a lightweight Python package fully compatible with JAX’s automatic differentiation framework. Our approach uniquely integrates Richardson extrapolation with Tikhonov regularized solutions computed across multiple nugget values, thereby preserving end-to-end differentiability while avoiding the information loss inherent in single-solution strategies. Experimental results demonstrate that autonugget significantly enhances solution accuracy and training stability without compromising rapid prototyping capabilities.

该研究针对参数ODE控制模型，提出一种计算可观测函数生成集的算法，通过Lie导数和可识别参数组合提高效率。

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 absence of readily available Gaussian quadrature rules for nonclassical weight functions by proposing a general framework that constructs such rules for arbitrary weights via the method of moments and the Stieltjes procedure. Innovatively integrating type-generic programming with adaptive high-precision arithmetic, the approach effectively controls round-off errors and, for the first time, systematically introduces tailored Gaussian quadrature methods to the statistics community. Implemented in Julia as the CustomGaussQuadrature package—accessible from R through JuliaConnectoR—the resulting quadrature rules achieve exact integration of polynomials up to degree \(2n-1\) while substantially reducing the number of function evaluations, thereby offering both high accuracy and computational efficiency.