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AutoNumerics: An Autonomous, PDE-Agnostic Multi-Agent Pipeline for Scientific Computing

Jianda Du, Youran Sun, Haizhao Yang

TL;DR

This work introduces AutoNumerics, a multi-agent framework that autonomously designs, implements, debugs, and verifies numerical solvers for general PDEs directly from natural language descriptions, and generates transparent solvers grounded in classical numerical analysis.

Abstract

PDEs are central to scientific and engineering modeling, yet designing accurate numerical solvers typically requires substantial mathematical expertise and manual tuning. Recent neural network-based approaches improve flexibility but often demand high computational cost and suffer from limited interpretability. We introduce \texttt{AutoNumerics}, a multi-agent framework that autonomously designs, implements, debugs, and verifies numerical solvers for general PDEs directly from natural language descriptions. Unlike black-box neural solvers, our framework generates transparent solvers grounded in classical numerical analysis. We introduce a coarse-to-fine execution strategy and a residual-based self-verification mechanism. Experiments on 24 canonical and real-world PDE problems demonstrate that \texttt{AutoNumerics} achieves competitive or superior accuracy compared to existing neural and LLM-based baselines, and correctly selects numerical schemes based on PDE structural properties, suggesting its viability as an accessible paradigm for automated PDE solving.

AutoNumerics: An Autonomous, PDE-Agnostic Multi-Agent Pipeline for Scientific Computing

TL;DR

This work introduces AutoNumerics, a multi-agent framework that autonomously designs, implements, debugs, and verifies numerical solvers for general PDEs directly from natural language descriptions, and generates transparent solvers grounded in classical numerical analysis.

Abstract

PDEs are central to scientific and engineering modeling, yet designing accurate numerical solvers typically requires substantial mathematical expertise and manual tuning. Recent neural network-based approaches improve flexibility but often demand high computational cost and suffer from limited interpretability. We introduce \texttt{AutoNumerics}, a multi-agent framework that autonomously designs, implements, debugs, and verifies numerical solvers for general PDEs directly from natural language descriptions. Unlike black-box neural solvers, our framework generates transparent solvers grounded in classical numerical analysis. We introduce a coarse-to-fine execution strategy and a residual-based self-verification mechanism. Experiments on 24 canonical and real-world PDE problems demonstrate that \texttt{AutoNumerics} achieves competitive or superior accuracy compared to existing neural and LLM-based baselines, and correctly selects numerical schemes based on PDE structural properties, suggesting its viability as an accessible paradigm for automated PDE solving.
Paper Structure (21 sections, 1 equation, 1 figure, 5 tables)

This paper contains 21 sections, 1 equation, 1 figure, 5 tables.

Table of Contents

  1. Introduction
  2. Position relative to prior work.
  3. Contributions.
  4. Method
  5. Problem Formulation and Plan Generation
  6. Coarse-to-Fine Execution
  7. Verification and Analysis
  8. Experiments & Results
  9. Experimental Setup
  10. Results and Analysis
  11. Conclusion
  12. Related Work
  13. Classical Numerical Methods.
  14. Neural and Data-Driven PDE Solvers.
  15. LLMs for Scientific Computing and PDE Automation.
  16. ...and 6 more sections

Figures (1)

  • Figure 1: The AutoNumerics pipeline. Steps 1--4 handle problem formulation and plan selection. Step 5 implements the coarse-to-fine execution strategy with Fresh Restart logic. Steps 6--7 perform verification and theoretical analysis.