# PDE-Agent: A toolchain-augmented multi-agent framework for PDE solving arXiv: https://arxiv.org/abs/2512.16214 ยท HTML: https://mlsanity.com/paper/2512.16214/ ## Key facts - Cited by 7 papers in the map (data through 14 Oct 2026). - Placed in Web, search, and research agents, one of 16 directions in LLM agents and tool use. - Builds on 32 of its 48 references that are in the map. - First posted on arXiv 18 Dec 2025. ## Placed in - LLM agents and tool use / [Web, search, and research agents](https://mlsanity.com/agents/web-search-and-research-agents/) ## Builds on (most cited first, 30 of 32) - [Language Models are Few-Shot Learners](https://mlsanity.com/paper/2005.14165/) (2020) - [GPT-4 Technical Report](https://mlsanity.com/paper/2303.08774/) (2023) - [ReAct: Synergizing Reasoning and Acting in Language Models](https://mlsanity.com/paper/2210.03629/) (2022) - [GPT-4o System Card](https://mlsanity.com/paper/2410.21276/) (2024) - [Toolformer: Language Models Can Teach Themselves to Use Tools](https://mlsanity.com/paper/2302.04761/) (2023) - [Gemini 2.5: Pushing the Frontier with Advanced Reasoning, Multimodality, Long Context, and Next Generation Agentic Capabilities](https://mlsanity.com/paper/2507.06261/) (2025) - [node2vec: Scalable Feature Learning for Networks](https://mlsanity.com/paper/1607.00653/) (2016) - [WebGPT: Browser-assisted question-answering with human feedback](https://mlsanity.com/paper/2112.09332/) (2021) - [When Not to Trust Language Models: Investigating Effectiveness of Parametric and Non-Parametric Memories](https://mlsanity.com/paper/2212.10511/) (2022) - [CRITIC: Large Language Models Can Self-Correct with Tool-Interactive Critiquing](https://mlsanity.com/paper/2305.11738/) (2023) - [DeepXDE: A deep learning library for solving differential equations](https://mlsanity.com/paper/1907.04502/) (2019) - [Scientific Machine Learning through Physics-Informed Neural Networks: Where we are and What's next](https://mlsanity.com/paper/2201.05624/) (2022) - [PDEBENCH: An Extensive Benchmark for Scientific Machine Learning](https://mlsanity.com/paper/2210.07182/) (2022) - [CodeAgent: Enhancing Code Generation with Tool-Integrated Agent Systems for Real-World Repo-level Coding Challenges](https://mlsanity.com/paper/2401.07339/) (2024) - [ToolQA: A Dataset for LLM Question Answering with External Tools](https://mlsanity.com/paper/2306.13304/) (2023) - [Tool Learning with Large Language Models: A Survey](https://mlsanity.com/paper/2405.17935/) (2024) - [Large Language Model Agent: A Survey on Methodology, Applications and Challenges](https://mlsanity.com/paper/2503.21460/) (2025) - [MDAgents: An Adaptive Collaboration of LLMs for Medical Decision-Making](https://mlsanity.com/paper/2404.15155/) (2024) - [Alita: Generalist Agent Enabling Scalable Agentic Reasoning with Minimal Predefinition and Maximal Self-Evolution](https://mlsanity.com/paper/2505.20286/) (2025) - [The Impact of Large Language Models on Scientific Discovery: a Preliminary Study using GPT-4](https://mlsanity.com/paper/2311.07361/) (2023) - [PINNacle: A Comprehensive Benchmark of Physics-Informed Neural Networks for Solving PDEs](https://mlsanity.com/paper/2306.08827/) (2023) - [Scientific Large Language Models: A Survey on Biological & Chemical Domains](https://mlsanity.com/paper/2401.14656/) (2024) - [AgentRxiv: Towards Collaborative Autonomous Research](https://mlsanity.com/paper/2503.18102/) (2025) - [OctoTools: An Agentic Framework with Extensible Tools for Complex Reasoning](https://mlsanity.com/paper/2502.11271/) (2025) - [Confucius: Iterative Tool Learning from Introspection Feedback by Easy-to-Difficult Curriculum](https://mlsanity.com/paper/2308.14034/) (2023) - [LLM With Tools: A Survey](https://mlsanity.com/paper/2409.18807/) (2024) - [What Are Tools Anyway? A Survey from the Language Model Perspective](https://mlsanity.com/paper/2403.15452/) (2024) - [A Physics-Informed Machine Learning Approach for Solving Heat Transfer Equation in Advanced Manufacturing and Engineering Applications](https://mlsanity.com/paper/2010.02011/) (2020) - [Evaluating and Improving Tool-Augmented Computation-Intensive Math Reasoning](https://mlsanity.com/paper/2306.02408/) (2023) - [Pre-Act: Multi-Step Planning and Reasoning Improves Acting in LLM Agents](https://mlsanity.com/paper/2505.09970/) (2025) ## Built on by (most cited first, 7 of 7) - [AutoNumerics: An Autonomous, PDE-Agnostic Multi-Agent Pipeline for Scientific Computing](https://mlsanity.com/paper/2602.17607/) (2026) - [GRAFT-ATHENA: Self-Improving Agentic Teams for Autonomous Discovery and Evolutionary Numerical Algorithms](https://mlsanity.com/paper/2605.11117/) (2026) - [OpInf-LLM: Parametric PDE Solving with LLMs via Operator Inference](https://mlsanity.com/paper/2602.01493/) (2026) - [Reinforcement Learning with Verifiable Physics: Post-training LLMs with Continuous Rewards](https://mlsanity.com/paper/2607.10474/) (2026) - [An Agentic Evaluation Framework for AI-Generated Scientific Code in PETSc](https://mlsanity.com/paper/2603.15976/) (2026) - [InfEngine: A Self-Verifying and Self-Optimizing Intelligent Engine for Infrared Radiation Computing](https://mlsanity.com/paper/2602.18985/) (2026) - [Large language models for partial differential equation workflows](https://mlsanity.com/paper/2608.03600/) (2026) ## Abstract (arXiv) Solving Partial Differential Equations (PDEs) is a cornerstone of engineering and scientific research. Traditional methods for PDE solving are cumbersome, relying on manual setup and domain expertise. While Physics-Informed Neural Network (PINNs) introduced end-to-end neural network-based solutions, and frameworks like DeepXDE further enhanced automation, these approaches still depend on expert knowledge and lack full autonomy. In this work, we frame PDE solving as tool invocation via LLM-driven agents and introduce PDE-Agent, the first toolchain-augmented multi-agent collaboration framework, inheriting the reasoning capacity of LLMs and the controllability of external tools and enabling automated PDE solving from natural language descriptions. PDE-Agent leverages the strengths of multi-agent and multi-tool collaboration through two key innovations: (1) A Prog-Act framework with graph memory for multi-agent collaboration, which enables effective dynamic planning and error correction via dual-loop mechanisms (localized fixes and global revisions). (2) A Resource-Pool integrated with a tool-parameter separation mechanism for multi-tool collaboration. This centralizes the management of runtime artifacts and resolves inter-tool dependency gaps in existing frameworks. To validate and evaluate this new paradigm for PDE solving , we develop PDE-Bench, a multi-type PDE Benchmark for agent-based tool collaborative solving, and propose multi-level metrics for assessing tool coordination. Evaluations verify that PDE-Agent exhibits superior applicability and performance in complex multi-step, cross-step dependent tasks. This new paradigm of toolchain-augmented multi-agent PDE solving will further advance future developments in automated scientific computing. Our source code and dataset will be made publicly available.