CodePDE: An Inference Framework for LLM-driven PDE Solver Generation

TL;DR

This paper introduces CodePDE, an inference framework that leverages large language models to generate PDE solvers, demonstrating promising performance and providing insights into their capabilities and limitations.

Contribution

It pioneers the use of LLMs for PDE solver generation through an inference framework and evaluates their reasoning, debugging, and self-refinement abilities.

Findings

LLMs can effectively generate PDE solvers with proper inference strategies

Trade-offs exist between solver reliability and complexity in LLM-generated solutions

Insights into failure modes and design principles for LLM-driven PDE solving agents

Abstract

Partial differential equations (PDEs) are fundamental to modeling physical systems, yet solving them remains a complex challenge. Traditional numerical solvers rely on expert knowledge to implement and are computationally expensive, while neural-network-based solvers require large training datasets and often lack interpretability. In this work, we frame PDE solving as a code generation task and introduce CodePDE, the first inference framework for generating PDE solvers using large language models (LLMs). With CodePDE, we present a thorough evaluation on critical capacities of LLM for PDE solving: reasoning, debugging, self-refinement, and test-time scaling. CodePDE shows that, with advanced inference-time algorithms and scaling strategies, LLMs can achieve strong performance across a range of representative PDE problems. We also identify novel insights into LLM-driven solver generation, such as trade-offs between solver reliability and sophistication, design principles for LLM-powered PDE solving agents, and failure modes for LLM on hard tasks. These insights offer guidance for building more capable and reliable LLM-based scientific engines.