numerical simulation

Scientific software selection frequently suffers from non-reproducible benchmarks due to multi-library, multi-metric evaluation and dynamic evolution—such as the introduction of new algorithms or modifications to test cases and evaluation criteria. This paper addresses numerical integration over arbitrary 2D/3D domains with implicit or parameterized boundaries (cut-cell quadrature), proposing the first automated benchmarking framework that systematically integrates CI/CD engineering practices into scientific computing workflows. The framework unifies GitHub Actions, Docker, Python-based scheduling, Jupyter-based report generation, and semantically versioned result archiving. It supports automated configuration, execution, visualization, and historical result comparison. It achieves >90% automation for benchmark tasks and regression detection; reduces integration time for new libraries or algorithms by 70%; and enables precise attribution of performance deviations to specific code commits. The framework significantly enhances reliability, reproducibility, and evolutionary adaptability in scientific software evaluation.

This work addresses the challenges of integrating sparse linear algebra libraries into scientific computing applications—such as computational fluid dynamics (CFD), power grid simulation, and cardiac electrophysiology—including poor maintainability, high cross-platform adaptation costs, and tight coupling between application code and low-level implementations. We propose a modular integration framework built upon Ginkgo, which achieves loose coupling via a unified abstract interface, explicit decoupling of algorithms from hardware backends, and runtime backend selection. From a software engineering perspective, the framework significantly reduces integration complexity while enhancing portability, testability, and long-term maintainability. Experimental evaluation demonstrates that the framework sustains high performance across heterogeneous platforms (CPU/GPU), shortens the hardware adaptation cycle, and enables efficient, sustainable multi-domain simulation.

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.

Empirical studies of meta-algorithms—such as algorithm selection, configuration, and scheduling—suffer from poor reproducibility and high bias risk due to excessive degrees of freedom in experimental design and fragmented community practices. Method: This paper introduces the first systematic integration of long-standing best practices from the COSEAL community across subfields, yielding a unified, dynamically evolving methodology framework spanning the entire experimental lifecycle: problem formulation → experimental design → execution → analysis → result presentation. Grounded in empirical methodology, rigorous experimental design, statistical standards, and principles of scientific communication, it emphasizes controlled variable management, benchmark standardization, and result transparency. Contribution/Results: The framework significantly reduces experimental bias, enhances cross-study comparability, and strengthens scientific rigor. It has been adopted for onboarding new researchers and informing journal review criteria.

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 work addresses a key limitation of conventional LLM-based PDE solvers, which implicitly embed numerical strategies within generated code, making pre-execution validation and post-failure correction challenging. To overcome this, the authors propose AutoPDE, the first framework to explicitly model solution strategies as revisable, decoupled objects separate from implementation code. AutoPDE employs a three-stage pipeline—PDE type identification, numerical method selection, and adaptive parameter tuning—augmented by low-overhead trial solves and a reusable skill library to construct and refine strategies prior to code generation. Evaluated on the PDE Agent Bench, AutoPDE achieves a 54.5% pass rate, outperforming the strongest baseline by 14.2 percentage points, thereby substantially improving both the reliability and interpretability of AI-driven PDE solving.

This work addresses the absence of a standardized benchmark for evaluating code generation targeting partial differential equation (PDE) solvers, particularly with respect to numerical accuracy, computational efficiency, and compatibility with mainstream finite element libraries. To bridge this gap, the authors introduce the first multi-metric, multi-library benchmark for PDE solver generation, comprising 645 structured instances spanning six mathematical problem types and eleven PDE classes. The benchmark supports three major finite element frameworks—DOLFINx, Firedrake, and deal.II—and incorporates a staged evaluation framework that holistically assesses code executability, numerical correctness, and performance. Experimental results demonstrate that while current large language models can produce executable code, their success rate drops substantially when stringent accuracy and efficiency requirements are imposed, thereby underscoring the necessity and effectiveness of the proposed benchmark in advancing reliable and efficient automated PDE solver generation.

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.

This work addresses the interoperability challenges arising from inconsistent interfaces among numerical solvers by proposing and implementing MaRDI—a standardized, open interface tailored for nonlinear optimization. Designed with a modular architecture, MaRDI establishes a generic solver adapter layer that enables seamless integration of diverse optimizers and embeds naturally within physics-informed neural network (PINN) training pipelines. Its efficacy is demonstrated through application to the viscous Burgers equation, where it substantially reduces the development overhead and benchmarking costs associated with solver-specific bindings. By abstracting low-level implementation details, MaRDI allows researchers to focus on core algorithmic innovation while significantly enhancing the efficiency and reproducibility of cross-solver experimentation.

Existing differentiable PDE solvers lack a unified benchmark to evaluate their practical performance in terms of gradient correctness, computational overhead, numerical stability, and ease of integration. This work proposes an extensible benchmarking framework that, through containerized encapsulation (Tesseract) and a standardized gradient API, supports cross-language interoperability and multiple automatic differentiation strategies. For the first time, it enables a systematic comparison of 14 differentiable solvers spanning fluid dynamics, structural mechanics, and heat transfer. Experiments reveal order-of-magnitude differences in computational cost and Jacobian condition numbers across solvers, yet all converge to similar optimal solutions. These findings indicate that real-world bottlenecks lie primarily in memory consumption, numerical stability, and compatibility—not in optimization capability.