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

2026-02-02

AI Summary

Current large language models (LLMs) often exhibit limited generalization and struggle to simultaneously achieve high accuracy and success rates when solving partial differential equations (PDEs) with unseen parameters or boundary conditions. This work proposes OpInf-LLM, a novel framework that uniquely integrates operator inference with LLMs to construct reduced-order models from minimal solution data and enables PDE-solving tasks to be specified via natural language. Evaluated across diverse PDE settings, the method achieves substantially higher execution success rates and numerical accuracy compared to existing LLM-based approaches. OpInf-LLM offers low computational overhead, strong generalization capabilities, and a unified interface, establishing an efficient new paradigm for solving parametric PDEs.

Technology Category

• Machine Learning: Large Multimodal Models (LMMs)
• Natural Language Processing: (Large) Language Models
• Search and Optimization: Learning to Search

Application Category

• User Modeling, Personalization and Recommendation: Large Language Models (LLM) for user modeling and recommendation
• Economics, Online Markets and Human Computation: Cost models of using LLMs in production systems
• Search and Retrieval-Augmented AI: Search Tool Learning with LLM: Teaching LLMs to invoke search and make use of retrieved information

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.

Problem

• PDE solving
• large language models
• operator inference
• generalization
• numerical accuracy

Innovation

• Operator Inference
• Large Language Models
• Parametric PDE Solving
• Reduced-Order Modeling
• Natural Language Interface