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Jianda Du

University of Maryland, College Park

Y

Youran Sun

University of Maryland, College Park

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Haizhao Yang

University of Maryland, College Park

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References (25)

[1]
PDE-Agent: A toolchain-augmented multi-agent framework for PDE solving
2025Jianming Liu, Ren Zhu et al.
[2]
AgenticSciML: Collaborative Multi-Agent Systems for Emergent Discovery in Scientific Machine Learning
2025Qile Jiang, G. Karniadakis
[3]
Automated Code Development for PDE Solvers Using Large Language Models
2025Haoyang Wu, Xinxin Zhang et al.
[4]
CodePDE: An Inference Framework for LLM-driven PDE Solver Generation
2025Shanda Li, Tanya Marwah et al.
[5]
AIDE: AI-Driven Exploration in the Space of Code
2025Zhengyao Jiang, Dominik Schmidt et al.
[6]
PDE-Controller: LLMs for Autoformalization and Reasoning of PDEs
2025Mauricio Soroco, Jialin Song et al.
[7]
On the Benefits of Memory for Modeling Time-Dependent PDEs
2024Ricardo Buitrago Ruiz, Tanya Marwah et al.
[8]
A Comprehensive Survey of Scientific Large Language Models and Their Applications in Scientific Discovery
2024Yu Zhang, Xiusi Chen et al.
[9]
LLM and Simulation as Bilevel Optimizers: A New Paradigm to Advance Physical Scientific Discovery
2024Pingchuan Ma, Tsun-Hsuan Wang et al.
[10]
UPS: Efficiently Building Foundation Models for PDE Solving via Cross-Modal Adaptation
2024Junhong Shen, Tanya Marwah et al.
[11]
Alias-Free Mamba Neural Operator
2024Jianwei Zheng, Wei Li et al.
[12]
Multiple Physics Pretraining for Spatiotemporal Surrogate Models
2024Michael McCabe, Bruno Régaldo-Saint Blancard et al.
[13]
Mathematical discoveries from program search with large language models
2023Bernardino Romera-Paredes, M. Barekatain et al.
[14]
MathCoder: Seamless Code Integration in LLMs for Enhanced Mathematical Reasoning
2023Ke Wang, Houxing Ren et al.
[15]
BioCoder: a benchmark for bioinformatics code generation with large language models
2023Xiangru Tang, Bill Qian et al.
[16]
Towards Foundation Models for Scientific Machine Learning: Characterizing Scaling and Transfer Behavior
2023Shashank Subramanian, P. Harrington et al.
[17]
ChemCrow: Augmenting large-language models with chemistry tools
2023Andrés M Bran, Sam Cox et al.
[18]
Message Passing Neural PDE Solvers
2022Johannes Brandstetter, Daniel E. Worrall et al.
[19]
Choose a Transformer: Fourier or Galerkin
2021Shuhao Cao
[20]
Fourier Neural Operator for Parametric Partial Differential Equations
2020Zong-Yi Li, Nikola B. Kovachki et al.

Showing 20 of 25 references

Founder's Pitch

"AutoNumerics automates the generation and verification of numerical solvers for PDEs, improving accessibility in scientific computing."

Scientific ComputingScore: 5View PDF ↗

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5

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Why It Matters

This research is significant as it offers a more accessible, transparent, and efficient method to solve PDEs, which are crucial in scientific and engineering domains. Without such a solution, developing numerical solvers requires significant expertise and manual effort, slowing down research and innovation.

Product Angle

To productize this, create a SaaS platform where users enter descriptions of their PDE problems and receive generated solver code. This platform could target universities and R&D departments.

Disruption

It could replace manual coding of numerical solvers by experts, as well as traditional software packages that require significant setup and expertise.

Product Opportunity

The market includes academic institutions, research labs, and industries reliant on complex simulations, such as aerospace or pharmaceuticals. These entities currently depend on manual solver development, which is resource-intensive.

Use Case Idea

A software platform that allows scientists and engineers to input natural language descriptions of PDE problems and receive ready-to-use numerical solvers, reducing the need for specialized programming skills.

Science

AutoNumerics leverages a multi-agent framework to generate numerical solvers for PDEs from natural language input. It includes a planning agent that proposes numerical schemes, a coarse-to-fine execution strategy to debug and verify code, and a residual-based self-verification method to ensure accuracy without analytical solutions.

Method & Eval

The system was tested on 24 PDE problems, comparing its performance against existing neural network and LLM-based methods. It achieved superior accuracy on these benchmarks, especially in choosing the appropriate numerical schemes based on PDE properties.

Caveats

There are limitations concerning high-dimensional and high-order PDEs as well as untested application in irregular domains. The system's reliance on a single LLM version might also constrain versatility or adaptability.

Author Intelligence

Jianda Du

University of Maryland, College Park
jdu37576@umd.edu

Youran Sun

University of Maryland, College Park
sun1245@umd.edu

Haizhao Yang

University of Maryland, College Park
hzyang@umd.edu