PDE-Agents: An LLM-Orchestrated Multi-Agent Framework for Automated Finite Element Simulations with Knowledge Graph-Augmented Reasoning
We present PDE-Agents, a multi-agent ecosystem that automates the full lifecycle of partial differential equation (PDE) and finite element method (FEM) simulations through natural-language interaction. Three large language model agents, Simulation, Analytics, and Database, are orchestrated by a LangGraph supervisor and run locally using Qwen3-Coder-Next and Llama 4 Scout on dual NVIDIA RTX PRO 6000 Blackwell GPUs. The model-agnostic architecture is validated across two generations of open-source LLMs. A GraphRAG knowledge base using Neo4j and 768-dimensional vector embeddings provides material properties, failure patterns, and prior run lineage. We report seven contributions: (i) verification and validation showing second-order spatial convergence, O(h^2), for three heat-equation benchmarks; (ii) a 50-task ablation with a frozen knowledge graph comparing KG On, KG Off, and KG Smart, where KG Smart achieves 100% success and the highest output quality, with physics score 0.933 versus 0.853 and material property fidelity (MPF) 0.926 versus 0.796 for KG Off; (iii) a novel-material study using three fictional materials known only to the knowledge graph, where KG Smart reaches MPF = 1.00 versus 0.34 without the graph; (iv) failure analysis tracing KG On's three failures to budget exhaustion and timeout, identifying warm-start injection as the main reliability factor; (v) an adaptive framework selecting retrieval mode per task; (vi) production metrics from 1,369 runs showing 97.8% overall success and 85.4% first-try success; and (vii) a 100-task knowledge-graph growth study showing an 8.8% MPF gain on hard tasks while easy and novel tasks remain at ceiling. All code, models, and evaluation artifacts are openly released. These results show that integration pattern, rather than knowledge content alone, determines whether GraphRAG helps or hinders LLM agents.