AutoPDE: Reliable Agentic PDE Solving via Explicitly Represented Solver Strategies

AutoPDE introduces a novel agentic approach to solving partial differential equations by maintaining solver strategies as explicit, inspectable objects rather than implicit code details. The system achieves a 54.5% pass rate on PDE Agent Bench, improving upon existing baselines by 14.2 percentage points through a three-stage process combining PDE analysis, numerical method selection, and adaptive tuning.

AutoPDE addresses a fundamental limitation in current LLM-based coding agents: their inability to separate numerical strategy from implementation details. Traditional approaches route feedback from failed solves directly to code modifications, making it difficult to identify whether failures stem from algorithmic choices or programming errors. This research demonstrates that explicitly representing solver strategies as independent objects creates a more debuggable and revisable workflow.

The development of reliable PDE solvers has long required domain expertise in numerical analysis, discretization methods, and stabilization techniques. Recent advances in LLM-based code generation promised to democratize this process by automating solver implementation, yet they left the critical strategic layer implicit. AutoPDE's innovation lies in maintaining a structured representation of numerical decisions—discretization choices, stabilization approaches, linear solver configuration, and resolution parameters—before generating any code. This allows engineers and researchers to validate fundamental decisions against PDE structure before computational investment.

The three-stage methodology reflects sound numerical analysis practice: first understanding the PDE's mathematical properties, then selecting appropriate numerical approaches, and finally calibrating parameters through pilot runs. The 54.5% success rate represents meaningful progress in automating complex scientific computing tasks. For researchers and engineers working with PDEs, this framework reduces both cognitive burden and computational waste by catching strategy-level errors early.

Looking forward, AutoPDE's approach of explicitly representing domain-specific strategies could influence broader AI-assisted engineering workflows beyond PDEs. The methodology demonstrates how separating high-level decision-making from implementation execution improves both transparency and debuggability in technical AI applications.

• →AutoPDE achieves 54.5% pass rate by maintaining explicit solver strategies separate from code implementation
• →Three-stage approach combines PDE analysis, numerical method selection, and adaptive parameter tuning before code generation
• →Explicit strategy representation enables numerical evidence-driven revisions when solves fail
• →Framework improves over strongest baseline by 14.2 percentage points on PDE Agent Bench
• →Methodology demonstrates how structured domain knowledge representation enhances LLM-based scientific computing