Large language models for partial differential equation workflows
cs.AI
Submitted: 2026-08-04
Updated: 2026-09-20
License: http://creativecommons.org/licenses/by/4.0/
The gist: Numerical methods and deep learning have advanced the modelling and simulation of systems governed by partial differential equations (PDEs), while formulating equations, configuring solvers, and
Terminology
Abstract
Numerical methods and deep learning have advanced the modelling and simulation of systems governed by partial differential equations (PDEs), while formulating equations, configuring solvers, and translating simulations into design and control decisions continue to demand substantial domain expertise and computational resources. Large language models (LLMs) offer opportunities to automate and coordinate these tasks by combining scientific knowledge, mathematical reasoning, code generation, and tool use. Here we review LLM-assisted PDE research across three stages, including discovery, solving, and optimization. We examine how LLMs interact with physical knowledge, numerical methods, and computational feedback to formulate models, implement solution procedures, and guide design and control. We also review existing benchmarks and evaluation practices for assessing scientific validity, computational performance, and workflow reliability. Finally, we discuss three interconnected challenges and opportunities: developing mathematical and physical reasoning tailored to specific PDE problems; building transferable domain expertise through domain-specific and multimodal post-training; and integrating active experimental design with scientific validation for open-ended problems. These directions outline a path towards a more substantive role for LLMs in developing and testing scientific ideas and computational methods across PDE research.
Sources
- MRKL Systems: A modular, neuro-symbolic architecture that combines large language models, external knowledge sources and discrete reasoning
- LLM4ED: Large Language Models for Automatic Equation Discovery
- PDE-SHARP: PDE Solver Hybrids through Analysis and Refinement Passes
- DrSR: LLM based Scientific Equation Discovery with Dual Reasoning from Data and Experience
- LLM-SR: Scientific Equation Discovery via Programming with Large Language Models
- From Equations to Insights: Unraveling Symbolic Structures in PDEs with LLMs
- LLM and Simulation as Bilevel Optimizers: A New Paradigm to Advance Physical Scientific Discovery
- PINNsAgent: Automated PDE Surrogation with Large Language Models
- Text-Trained LLMs Can Zero-Shot Extrapolate PDE Dynamics, Revealing a Three-Stage In-Context Learning Mechanism
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