applied research

Existing LLM mathematical benchmarks emphasize exact solutions or formal proofs, overlooking pervasive approximate modeling tasks in applied sciences. Method: We introduce HARDMath2—a high-quality, expert-curated benchmark focused on asymptotic analysis and applied mathematics—comprising 211 original problems spanning boundary layer theory, the WKB method, and asymptotic solutions to nonlinear PDEs. Developed collaboratively by Harvard faculty and students, it employs a novel “student-led, human–model interactive” construction paradigm: difficult problems are reverse-engineered from LLM failure cases and refined via human authoring, peer verification, automated LLM solving, numerical solution validation, and asymptotic modeling verification. Contribution/Results: State-of-the-art LLMs perform poorly on HARDMath2, revealing critical gaps in asymptotic reasoning. Notably, students deepened their own mathematical understanding by diagnosing model errors. HARDMath2 fills a fundamental gap in evaluating LLMs’ applied mathematical competence and establishes a new paradigm for rigorous, pedagogically informed reasoning assessment.

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 study addresses the widespread challenge in engineering of solving transcendental equations that typically require iterative numerical methods. It systematically evaluates the performance of large language models (LLMs) in both direct numerical prediction and hybrid symbolic–numerical approaches. The authors propose a novel hybrid solving paradigm that integrates LLM-based symbolic manipulation, initial value estimation, and the Newton–Raphson method, evaluated across 100 problems spanning seven engineering domains. Results demonstrate that LLMs are better suited as intelligent interfaces for symbolic operations rather than high-precision numerical computation. The hybrid approach reduces average relative error by 67.9%–81.8%, with improvements reaching up to 93.1% in electronics, significantly outperforming pure LLM-based prediction and confirming its effectiveness and broad applicability in engineering computation.

This work addresses the challenge of solving complex mathematical optimization problems through human–AI collaboration. Method: We propose AlphaEvolve—a novel autonomous coding agent that synergistically integrates large language models (LLMs) with evolutionary algorithms. Operating via iterative “generate–evaluate–mutate–select” cycles, AlphaEvolve automatically discovers mathematical constructions, induces general formulas, and collaborates with DeepThink and AlphaProof for automated formal proof and deep deductive reasoning. Crucially, it unifies symbolic computation and formal verification within an LLM-guided evolutionary search framework. Contribution/Results: Evaluated on 67 open mathematical problems spanning diverse domains—including combinatorics, number theory, and discrete optimization—AlphaEvolve reproduces most known optimal solutions and achieves breakthrough results on several previously unresolved instances. The framework significantly reduces both manual intervention and computational overhead, establishing a new paradigm for AI–mathematician co-reasoning in advanced mathematical discovery.

Doctoral students in life sciences commonly lack formal software engineering training, hindering the development of robust, reproducible, and collaborative research software. Method: This study proposes ten pedagogical principles for research software development, establishing the first systematic framework centered on “research software pedagogy”—distinct from generic programming instruction. It integrates software engineering best practices (e.g., Git-based version control, CI/CD pipelines, unit testing, RESTful API design), learning science principles, and authentic research workflows, emphasizing the seamless embedding of automation, documentation, testing, and collaborative practices throughout the research lifecycle. Contribution/Results: The framework delivers a generalizable, plug-and-play pedagogical paradigm. Deployed across multiple Chinese universities’ life sciences PhD programs, it has demonstrably improved software deliverable quality, code reusability, and cross-team collaboration efficiency—bridging critical gaps between computational literacy and rigorous, team-based scientific software practice.

This work addresses a critical gap in existing mathematical formalization benchmarks, which predominantly focus on propositional verification while neglecting the evaluation of explicit solution construction—such as numerical values or algorithms—particularly in applied mathematics. To bridge this gap, the authors propose a construct-and-verify workflow framework that requires agents to first generate concrete solutions and then formally prove their correctness. Building upon this framework, they introduce AMBER, a novel benchmark for applied mathematical reasoning spanning convex analysis, optimization, numerical linear algebra, and high-dimensional probability. Implemented in Lean 4, this benchmark enables the first systematic evaluation of large language models on constructive tasks, revealing that general-purpose reasoning models significantly outperform specialized theorem provers, the latter suffering from “tactic overfitting” that limits their generalization. The study further underscores the pivotal role of instruction-following capability in multi-task formal reasoning.

This work investigates the creative space of mathematical proofs under constraints, with a particular focus on the impact of non-constructive reasoning. We introduce a strategy ablation methodology that integrates our custom-built Meno automated formalization tool with Goedel Prover embeddings to systematically explore both formal and informal proof spaces for foundational theorems from *Analysis I* within the Lean theorem prover. Our experiments successfully generate a novel class of machine-produced proofs, revealing that these proofs cluster along low-dimensional submanifolds in a high-dimensional representation space and significantly diverge from human-constructed proof trajectories. This study provides the first quantitative characterization of the structural differences between machine-generated and human proofs.

Existing formal mathematical benchmarks predominantly focus on Olympiad-style problems and algebra, with limited coverage of computational and applied mathematics. This work introduces CAM-Bench, the first systematic benchmark comprising 1,000 Lean 4 proof goals derived from exercises in classical textbooks, spanning optimization, numerical linear algebra, and numerical analysis. Through a pipeline involving dependency recovery, context normalization, formal translation, and semantic alignment verification, the original problems are transformed into self-contained theorems with complete contextual information. CAM-Bench not only fills a critical gap in the landscape of formal mathematical reasoning benchmarks but also exposes characteristic failure modes of large language models, particularly in handling local assumptions, invoking foundational theorems, and conducting long-horizon logical reasoning.

This work addresses the lack of a unified, open-source, and modular platform for collaboratively exploring shape and topology optimization methods in both teaching and research. The authors present an object-oriented, MATLAB-based open-source framework that employs abstract base classes to define core interfaces, enabling seamless integration of parametric and level-set-based shape optimization alongside density-based, level-set, and topological sensitivity approaches to topology optimization. By directly mapping mathematical formulations to executable code, the framework allows users to extend objective functionals or constraints simply by deriving new classes without modifying the core implementation. Highly modular and reproducible, the framework bridges the gap between shape and topology optimization, offering a continuous research pathway. Its effectiveness and flexibility are demonstrated through diverse numerical examples in both educational and research contexts.

This study investigates whether artificial intelligence can effectively contribute to creative mathematical research under rigorous human supervision. By establishing a human–AI collaborative framework that integrates symbolic algebra manipulation, automated proof exploration, semantic synthesis of mathematical literature, and LaTeX-based formalization—augmented by human mathematical intuition and verification—the work systematically discovers and proves novel error representations and bounds for Hermite quadrature formulas. The research presents the first fully documented, high-transparency account of an end-to-end human–AI co-discovery process in mathematics, elucidating effective collaboration patterns and failure modes. It thereby establishes a viable pathway and validation protocol for AI-assisted mathematical inquiry, extends classical error theory, and underscores the irreplaceable role of human domain expertise in advanced mathematical discovery.