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

TL;DR

OpInf-LLM addresses the challenge of solving diverse PDEs with LLMs by integrating operator inference-based reduced-order models with language-driven task specification. The offline stage learns a shared POD basis and parameter-dependent reduced operators, transforming PDE generalization into polynomial fitting and low-dimensional ODE integration, while the online stage leverages LLMs to parse natural language tasks and perform ROM inference and time integration via tool calls. Across heat, Burgers, and cavity PDEs, OpInf-LLM achieves high execution success and accurate predictions for unseen parameters and boundary conditions, outperforming CodePDE, MOL-LLM, and direct LLM solvers in robustness and efficiency. This framework offers a scalable path toward generalizable, language-enabled reduced-order PDE solvers with improved reliability and interpretability for scientific computing.

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 based on operator inference. The proposed framework leverages a small amount 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 specification of PDE solving tasks. Its low computational demands and unified tool interface further enable a high execution success rate across heterogeneous settings. By combining operator inference with LLM capabilities, OpInf-LLM opens new possibilities for generalizable reduced-order modeling in LLM-based PDE solving.

Figures (9)

• Figure 1: Trade-offs in prediction error and success rate.
• Figure 2: Overall diagram of the OpInf-LLM framework.
• Figure 3: Heat equation predictions for $\nu=1.0$ and $\nu=3.0$. All plots share the same color scale. Results are shown for CodePDE, LLM and OpInf-LLM with GPT-4.1.
• Figure 4: Burgers' equation predictions for $\nu=0.03$ and $\nu=0.07$. All plots share the same color scale. Results are shown for CodePDE, LLM and OpInf-LLM with GPT-4.1.
• Figure 5: Lid-driven cavity predictions for $\mathrm{Re}=80$ and $\mathrm{Re}=120$ at $t = 1$ and $t = 2$. All plots share the same color scale. Results are shown for CodePDE, LLM and OpInf-LLM with GPT-4.1.