Verifier-guided discovery of exact high-order mimetic operators with large language models

Abstract

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.