OpInf-LLM: Parametric PDE Solving with LLMs via Operator Inference

TL;DR

OpInf-LLM introduces a framework that combines operator inference with large language models to accurately and efficiently solve diverse PDEs, including unseen cases, with high success rates.

Contribution

It presents a novel LLM-based PDE solving method leveraging operator inference, enabling generalization to unseen parameters with low computational costs.

Findings

Achieves high success rate across heterogeneous PDE settings.

Accurately predicts solutions for unseen parameters and configurations.

Provides a unified, low-cost solution pipeline for PDE solving.

Abstract

Solving diverse partial differential equations (PDEs) is fundamental in science and engineering. Large language models (LLMs) have demonstrated strong capabilities in code generation, symbolic reasoning, and tool use, but reliably solving PDEs across heterogeneous settings remains challenging. Prior work on LLM-based code generation and transformer-based foundation models for PDE learning has shown promising advances. However, a persistent trade-off between execution success rate and numerical accuracy arises, particularly when generalization to unseen parameters and boundary conditions is required. In this work, we propose OpInf-LLM, an LLM parametric PDE solving framework via operator inference. The proposed framework leverages small amounts of solution data to enable accurate prediction of diverse PDE instances, including unseen parameters and configurations, and provides seamless integration with LLMs for natural language task specification and physics-based reasoning of proper feature parameterization. Its low computational demands and unified solution pipeline further enable a high execution success rate across heterogeneous settings, opening new possibilities for generalizable reduced-order modeling in LLM-based PDE solving.