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

TL;DR

AutoNumerics is an autonomous multi-agent system that designs, implements, and verifies classical numerical PDE solvers from natural language, combining interpretability with competitive accuracy.

Contribution

It introduces a novel multi-agent framework that automates PDE solver development from natural language, integrating classical analysis with AI techniques.

Findings

Achieves competitive or superior accuracy on 24 PDE problems.

Correctly selects numerical schemes based on PDE properties.

Demonstrates viability as an accessible automated PDE solving paradigm.

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.