PDE-Agent: A toolchain-augmented multi-agent framework for PDE solving

TL;DR

This work introduces PDE-Agent, a toolchain-augmented multi-agent framework that turns natural language PDE problems into automated, multi-tool solving workflows. It combines Prog-Act, a dual-loop error handling mechanism with Graph Memory, and a Resource-Pool for implicit-parameter coordination to enable robust, autonomous PDE solving. A new PDE-Bench and multi-level evaluation metrics assess global success, tool-usage quality, and logical collaboration, with experiments showing superior performance over baselines. The approach advances automated scientific computing by improving autonomy, reliability, and interpretability in PDE solving, and code and data will be publicly released.

Abstract

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.

Figures (3)

• Figure 1: The competitive and stable performance across diverse PDE problems.
• Figure 2: The architecture of PDE-Agent.
• Figure 3: The multi-level evaluation.