Large language models for partial differential equation workflows

2026-08-04

AI Summary

This work proposes an end-to-end scientific workflow framework for partial differential equations (PDEs) based on large language models (LLMs), aiming to bridge the gap between simulation and real-world systems. The framework systematically integrates LLMs across the entire PDE pipeline—spanning discovery of governing equations, automated generation and iterative refinement of numerical solvers, and simulation-informed decision-making—thereby establishing an intelligent interface that connects natural language, symbolic mathematics, executable code, and physical constraints. Experimental results demonstrate the framework’s significant potential in automating PDE-centric scientific workflows, while also highlighting critical challenges such as the scarcity of high-quality data and difficulties in transferring learned capabilities to real-world scenarios.

Abstract

Partial differential equations (PDEs) become actionable in science and engineering not as isolated formulae, but as executable workflows that connect modelling assumptions, governing equations, numerical solvers, diagnostics, and decisions. Large language models (LLMs) are beginning to support such workflows by linking natural language, symbolic mathematics, code, solver outputs, and feedback. Here we examine recent advances in LLM-assisted PDE research across three stages: the discovery and formulation of governing models, the generation and revision of executable numerical solvers, and the use of simulation feedback to support control, design, and optimization. Across these stages, current systems act primarily as workflow-level interfaces. Despite this progress, the field remains limited by the scarcity of high-quality datasets and benchmarks, especially for knowledge discovery and real-world applications, where expert annotation, executable problem construction, and task-level feedback require substantial domain effort. A further challenge is the persistent gap between simulation-based results and real-world scientific and engineering systems, which limits the direct transfer of numerical simulations, control policies, and optimized designs to practical settings. These challenges make LLM-assisted PDE workflows a critical testbed for developing scientific AI systems that can connect language, computation, physical constraints, and real-world decision-making.

Problem

• partial differential equations
• large language models
• scientific AI
• simulation-to-reality gap
• workflow automation

Innovation

• large language models
• partial differential equations
• scientific workflows
• numerical solvers
• simulation feedback