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

Shanda Li, Tanya Marwah, Junhong Shen, Weiwei Sun, Andrej Risteski, Yiming Yang, Ameet Talwalkar · May 13, 2025 · Citations: 0

Automatic Metrics Coding

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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.

Human Data Lens

Uses human feedback: No
Feedback types: None
Rater population: Domain Experts
Unit of annotation: Unknown
Expertise required: Coding

Evaluation Lens

Evaluation modes: Automatic Metrics
Agentic eval: None
Quality controls: Not reported
Confidence: 0.30
Flags: low_signal, possible_false_positive

Research Summary

Contribution Summary

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).

Why It Matters For Eval

With CodePDE, we present a thorough evaluation on critical capacities of LLM for PDE solving: reasoning, debugging, self-refinement, and test-time scaling.
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.