Automated Code Development for PDE Solvers Using Large Language Models

Foundation models -- large language models (LLMs) in particular -- have become ubiquitous, shaping daily life and driving breakthroughs across science, engineering, and technology. Harnessing their broad cross-domain knowledge, text-processing, and reasoning abilities for software development, e.g., numerical libraries for solving partial differential equations (PDEs), is therefore attracting growing interest. Yet existing studies mainly automate case setup and execution for end users. We introduce LLM-PDEveloper, a zero-shot, multi-agent LLM framework that automates code development for PDE libraries, specifically targeting secondary developers. By translating mathematical and algorithmic descriptions directly into source code, LLM-PDEveloper generates new solvers/modules and adapts existing ones. This end-to-end math-to-code approach enables a self-augmenting pipeline that continuously expands the codebase of a library, extends its capacities, and broadens its scope. We demonstrate LLM-PDEveloper on three tasks: 1) build a solver for a new PDE, 2) implement new BCs for a given PDE, and 3) modify an existing solver to incorporate additional terms, achieving moderate success rates. Failures due to syntactic errors made by LLMs are analyzed and we propose effective fixes. We also identify the mechanisms underlying certain semantic errors, guiding future research.