discretize continuous models and equations

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 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.

Existing discretization methods for continuous-time linear systems separately approximate the dynamics, input, and process noise matrices—leading to error accumulation and computational inconsistency. To address this, we propose a unified analytical discretization method based on a single matrix exponential computation. Our approach achieves the first joint closed-form discretization of all three state-space matrices (A, B, G), eliminating errors inherent in conventional zero-order-hold (ZOH) discretization that relies on multi-step numerical integration or component-wise approximations. Efficient matrix exponential evaluation is performed via the scaling-and-squaring method combined with Padé approximation. Experimental validation on canonical LTI systems confirms strict equivalence to exact ZOH discretization, with discretization error reduced by one to two orders of magnitude and computational time decreased by approximately 40%. The method thus significantly enhances accuracy, numerical consistency, and computational efficiency.

This work addresses the problem of deriving provably tight floating-point rounding error bounds for numerical programs featuring conditional branches, no loops, and mixed-precision arithmetic. Methodologically, it unifies the modeling of conditional control flow and precision heterogeneity via two novel quantitative metrics—“instability jumps” and “window width”—and integrates interval arithmetic, abstract interpretation, and precision-aware semantic modeling, augmented with abstraction-guided global optimization. Its key contribution is the first formal framework enabling joint, compositional analysis of conditional branching and mixed precision, achieving both high bound tightness and practical analysis efficiency. Experimental evaluation on standard benchmarks demonstrates significantly tighter error bounds compared to prior approaches. Furthermore, the framework successfully guides precision configuration—e.g., step size and search direction—in the conjugate gradient method, empirically validating its utility in supporting design-time trade-offs among accuracy, error bounds, and computational efficiency.

This work addresses the limitations of conventional physics-informed neural networks in solving partial differential equations, which often lack rigorous control over numerical errors and guarantees of robustness and convergence. The authors propose a novel framework based on a discrete variational formulation, wherein function spaces, inner products, and weak forms are defined over discrete point sets. By integrating neural networks with discrete automatic differentiation and employing Kronecker delta test functions, they construct a robust loss functional directly linked to the true approximation error. The method synergistically combines discrete finite-difference derivatives, automatic differentiation, and Adamax optimization to achieve controllable error behavior during training. Numerical experiments demonstrate that the approach exhibits superior convergence, robustness, and effective suppression of numerical errors on benchmark problems, including two-dimensional Stokes and Laplace equations.

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 the unclear relationship between continuous theory and discrete implementation in neural operators for solving partial differential equations, particularly concerning stability and discretization error. For the first time, rigorous discretization error bounds are established for State-Space Neural Operators (SS-NOs) and Fourier Neural Operators (FNOs). By leveraging functional analysis, the regularity of solutions is explicitly linked to input discretization, and Input-to-State Stability (ISS) theory is introduced to quantify how discretization affects stability in the continuous domain. Numerical experiments on one- and two-dimensional benchmark problems validate the tightness of the derived theoretical bounds, demonstrating that SS-NOs exhibit both robustness and numerical stability across varying resolutions.

This study addresses the lack of a rigorous formal definition of "finite elements" in the finite element method by proposing a formal framework within the Rocq proof assistant based on record types, wherein a finite element is modeled as a structure comprising geometric data and validity proofs. The work presents the first complete formalization of simplicial Lagrange finite elements of arbitrary dimension and polynomial degree in a proof assistant, and rigorously verifies their unisolvence property using foundational theories of finite families, affine spaces, and multivariate polynomials. This achievement yields a general definition and correctness proof for simplicial Lagrange finite elements with uniform nodal distributions, thereby establishing a formal foundation for the verification of scientific computing software.

Modern computational fluid dynamics (CFD) urgently requires seamless integration of simulation into design, optimization, and data-driven workflows, confronting challenges in the co-design of physical models, numerical methods, heterogeneous hardware, and automatic differentiation. This work systematically evaluates the suitability of the Julia programming language for CFD, leveraging its unified language ecosystem, multiple dispatch, and type specialization to deeply integrate high performance, differentiability, and software composability. Empirical validation through distributed CPU/multi-GPU parallelism, performance-portable frameworks, and open-source CFD projects demonstrates the feasibility of native Julia-based CFD at scale and its advantages in differentiable workflows. Nevertheless, the maturity of Julia’s industrial toolchain still lags behind that of conventional languages.