efficient numerical solvers

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 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.

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 high sensitivity of constraint programming solver performance to hyperparameter configurations and the prohibitive cost of manual tuning. The authors propose a resource-aware, two-phase auto-tuning framework that, within a limited time budget, first explores promising configurations and then solves the target problem using the best identified configuration. Innovatively integrating Bayesian optimization with Hamming distance-based search within a unified framework, the approach is implemented using CPMpy. Experimental evaluation on 114 combinatorial optimization instances demonstrates that the method outperforms the default configurations on 25.4% and 38.6% of instances for the ACE and Choco solvers, respectively, significantly surpassing either search strategy in isolation.

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 lack of interoperability among nonlinear optimization solvers in scientific computing, which typically necessitates extensive code refactoring and revalidation when switching solvers or invoking them across programming languages. To overcome this limitation, the authors propose a universal interface framework tailored for nonlinear optimization problems. The framework employs a modular architecture that integrates multi-language bindings and an automated data marshaling mechanism, enabling plug-and-play solver integration and seamless cross-language invocation. By abstracting solver-specific implementation details, the framework substantially reduces the development and verification overhead associated with solver substitution and cross-language collaboration, thereby significantly enhancing the iteration efficiency of scientific computing workflows.

This work proposes and implements a high-performance, unified numerical optimization framework in Rust, addressing the pervasive challenge of numerical optimization in scientific computing and engineering. The framework natively supports diverse constraint types, integrates multiple optimization algorithms, and offers a consistent, extensible interface design. As the first numerical optimization library in the Rust ecosystem to combine a rich suite of solvers, first-class constraint handling, and high computational efficiency, it demonstrates exceptional practicality and scalability across applications such as model fitting, simulation calibration, and machine learning training.

This work addresses the challenge that effective neural PDE solvers are extremely sparse in the design space, rendering large language models inefficient for their automated discovery. To overcome this, the authors propose ADSL-PDE, a framework that introduces structured search states to decouple high-level design decisions—such as architecture, physical constraints, and optimization objectives—from low-level code implementation. By formulating a domain-specific language to restructure the search space, the method substantially increases the density of valid candidate solvers. Building upon this representation, the framework integrates deterministic compiler-based mapping with an evolutionary algorithm guided by empirical feedback and large language model suggestions. Experiments across multiple PDE benchmarks demonstrate that the approach improves solver performance by over 52% within the first ten iterations, while significantly enhancing search efficiency and optimization stability.

This work proposes a multi-agent collaborative framework that automatically translates natural language descriptions of operations research problems into solvable mathematical models and executable code. To address common modeling challenges—such as semantic misinterpretation, structural flaws, and mathematical inconsistencies—the approach employs specialized agents to extract decision variables and constraints, integrating structured information extraction, iterative self-correction, and a fourfold feedback validation mechanism to achieve end-to-end modeling. Its modular architecture enhances transparency and auditability throughout the modeling process. Evaluated on four standard benchmarks encompassing linear programming (LP), mixed-integer linear programming (MILP), and nonlinear programming, the method achieves state-of-the-art performance on three and demonstrates highly competitive results on the fourth.