基于大语言模型的验证器引导式高阶精确仿射算子发现
Verifier-guided discovery of exact high-order mimetic operators with large language models
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中文总结 AI 辅助
本研究利用大语言模型提出构造程序，结合确定性验证器，发现精确高阶仿射算子，显著降低谱半径常数和PDE误差，并保证能量收缩性。
中文摘要 AI 辅助
设计高阶保结构离散化是一个受约束的数学搜索问题：守恒性、正离散内积、物理谱行为、边界精度、带宽以及偏微分方程（PDE）误差必须同时满足。我们测试大语言模型（LLMs）能否在保持非权威性的前提下提供帮助。作为动机的Corbino-Castillo交错算子的MOLE实现满足一般离散高斯恒等式并精确守恒，但其六阶和八阶Dirichlet块产生了四个非实边界局域模态。若采用端点支持的正定恒等式，则将强制实非正谱，因此搜索改变了闭包和范数架构。LLM仅提出类型化的构造程序；确定性线性规划编译器生成系数；独立验证器测试代数、正性、物理模态、条件数以及制造PDE；耦合有理重构提供精确证书。在1200次求解器评估中，外部固定验证器接受了55.0%的全度量反馈提议和53.3%的照明档案提议，而均匀随机搜索仅为13.3%。四个领先的LLM起源程序被精确重构。最强的六阶内部、四阶边界候选方案将先前的正对角谱半径常数降低了25.4%，其PDE误差降低了62.7倍。其认证的热方程能量是收缩的，而六阶参考表现出6.6%的瞬态增长；两者具有相同的RK4稳定性极限。LLM提出结构假设；确定性数学决定有效性。
英文摘要
Designing a high-order structure-preserving discretization is a constrained mathematical search: conservation, a positive discrete inner product, physical spectral behavior, boundary accuracy, bandwidth, and partial differential equation (PDE) error must hold simultaneously. We test whether large language models (LLMs) can help while remaining non-authoritative. The motivating MOLE implementation of the Corbino-Castillo staggered operators satisfies a general discrete Gauss identity and conserves exactly, yet its order-six and order-eight Dirichlet blocks develop four non-real boundary-localized modes. An endpoint-supported positive-definite identity would instead force a real non-positive spectrum, so the search changes the closure and norm architecture. An LLM proposes only a typed construction program; a deterministic linear-program compiler generates coefficients; an independent verifier tests algebra, positivity, physical modes, conditioning, and manufactured PDEs; and coupled rational reconstruction provides exact certificates. Across 1,200 solver evaluations, an externally fixed verifier accepted 55.0% of full-metric-feedback proposals and 53.3% of illumination-archive proposals, versus 13.3% for uniform random search. Four leading LLM-originated programs were reconstructed exactly. The strongest order-six-interior, order-four-boundary candidate lowers the prior positive-diagonal spectral-radius constant by 25.4% and its PDE error 62.7-fold. Its certified heat-equation energy is contractive, whereas the order-six reference exhibits 6.6% transient growth; both have the same RK4 stability limit. The LLM proposes structural hypotheses; deterministic mathematics determines validity.
