# AutoPDE: Reliable Agentic PDE Solving via Explicitly Represented Solver Strategies

arXiv: https://arxiv.org/abs/2606.10752 · HTML: https://mlsanity.com/paper/2606.10752/

## Key facts

- Cited by 3 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 9 of its 31 references that are in the map.
- First posted on arXiv 9 Jun 2026.

## 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, 9 of 9)

- [Self-Refine: Iterative Refinement with Self-Feedback](https://mlsanity.com/paper/2303.17651/) (2023)
- [MetaGPT: Meta Programming for A Multi-Agent Collaborative Framework](https://mlsanity.com/paper/2308.00352/) (2023)
- [SWE-agent: Agent-Computer Interfaces Enable Automated Software Engineering](https://mlsanity.com/paper/2405.15793/) (2024)
- [OpenHands: An Open Platform for AI Software Developers as Generalist Agents](https://mlsanity.com/paper/2407.16741/) (2024)
- [Teaching Large Language Models to Self-Debug](https://mlsanity.com/paper/2304.05128/) (2023)
- [CodePDE: An Inference Framework for LLM-driven PDE Solver Generation](https://mlsanity.com/paper/2505.08783/) (2025)
- [PDE-Controller: LLMs for Autoformalization and Reasoning of PDEs](https://mlsanity.com/paper/2502.00963/) (2025)
- [AutoNumerics: An Autonomous, PDE-Agnostic Multi-Agent Pipeline for Scientific Computing](https://mlsanity.com/paper/2602.17607/) (2026)
- [PDE-SHARP: PDE Solver Hybrids through Analysis and Refinement Passes](https://mlsanity.com/paper/2511.00183/) (2025)

## Built on by (most cited first, 3 of 3)

- [Large language models for partial differential equation workflows](https://mlsanity.com/paper/2608.03600/) (2026)
- [AutoPDEBench: Benchmarking LLM Auto-Research for Neural PDE Solver Design](https://mlsanity.com/paper/2609.32245/) (2026)
- [GPUPhysBench: Benchmarking Coding Agents for Correct and Efficient GPU Physics Simulation](https://mlsanity.com/paper/2609.35639/) (2026)

## Abstract (arXiv)

Numerical solvers for partial differential equations (PDEs) are core computational tools in science and engineering. Building reliable PDE solvers requires not only executable code, but a numerical solver strategy, a set of decisions about discretization, stabilization, solver configuration, and resolution control, that matches the PDE structure. Recent LLM-based coding agents have begun to reduce the programming burden by generating and debugging solver implementations. However, they typically move directly from a PDE problem to solver code, leaving the solver strategy implicit in implementation details. Feedback from a failed solve is therefore routed back to code edits rather than to the underlying strategy, so numerical decisions remain hard to check before code is generated and hard to revise using numerical evidence when it fails. To address this limitation, we propose AutoPDE, a code agent that maintains the solver strategy as an explicitly represented object throughout the solving process: an independent, inspectable object that is built before any code is written and can be revised, using numerical evidence, whenever a solve fails. AutoPDE builds and maintains this object in three stages, all drawing from a library of reusable PDE-solving skills: PDE analysis identifies the equation type and algebraic structure; numerical method selection chooses a numerical method that matches the analysis result and commits to a discretization, stabilization, and linear solver accordingly; and adaptive tuning runs low-cost pilot solves to calibrate resolution and tolerances under the prescribed accuracy and runtime budget. We evaluate AutoPDE on the PDE Agent Bench, where experimental results show that AutoPDE achieves a pass rate of 54.5%, improving over the strongest baseline by 14.2 percentage points.