MetaColloc: Optimization-Free PDE Solving via Meta-Learned Basis Functions

     

Paper | Hugging Face

This repository contains the official implementation of MetaColloc, an Optimization-Free PDE Solver via Meta-Learned Basis Functions.

💡 What is MetaColloc?

Traditional physical information neural networks (PINNs) usually regard the solution of partial differential equations (PDE) as an instance-specific optimization problem. This means that for every new equation or boundary condition, thousands of steps of time-consuming gradient descent need to be performed from scratch.

MetaColloc changes this. We propose to completely decouple "Basis Discovery" and "PDE Solving":

1. Offline Meta-Training: We offline train a two-branch neural network on multi-scale Gaussian random fields (GRF) to build a highly expressive universal neural basis dictionary.
2. Online Test-Time Solving: While testing, network parameters are completely frozen. Solving the new PDE becomes a pure closed-form linear algebra operation (for linear PDEs) or a fast Newton-Raphson iteration (for nonlinear PDEs).

⚙️ How it Works

1. Dual-Branch Architecture: Our network structure cleverly combines the "low-frequency raw coordinate branch" that processes smooth macroscopic signals and the "multi-scale Fourier Features (high-frequency branch)" that captures violent oscillations.
2. Collocation Matrix Assembly: When faced with a new domain, just randomly scatter points (Collocation points) and use forward-mode AutoDiff to extract $\Phi_x, \Phi_{xx}, ...$, and you can instantly assemble the linear equation system $A w = b$ and solve for the coefficient $w$.

📄 License

This project is licensed under the Apache-2.0 License.

📰 News

• [5/22/2026] Code and experiments released
• [5/12/2026] Preprint available on arXiv

🚀 Quick Start

🧪 Configure environment

mamba create -n metacolloc python=3.12.12
mamba activate metacolloc
mamba install --file requirements.txt

📧 Contact

Authors:

• Zichuan Yang (2153747@tongji.edu.cn)

Questions? Open an issue or email us!

📖 Citation

If you use this code in your research, please cite:

@misc{yang2026metacollocoptimizationfreepdesolving,
      title={MetaColloc: Optimization-Free PDE Solving via Meta-Learned Basis Functions}, 
      author={Zichuan Yang},
      year={2026},
      eprint={2605.12368},
      archivePrefix={arXiv},
      primaryClass={cs.LG},
      url={https://arxiv.org/abs/2605.12368}, 
}