whucsu/RBF-PINN

RBF-PINN: Improve your PINN with a single Feature Mapping Layer.

This is the official repo for the paper Feature Mapping in Physics-Informed Neural Networks (PINNs).
Preprint, Under Review
Chengxi Zeng, Tilo Burghardtg, Alberto M Gambaruto

Short paper: RBF-PINN: Non-Fourier Positional Embedding in Physics-Informed Neural Networks.
International Conference on Learning Representations (ICLR 2024), AI4DifferentialEquations in Science Workshop

Abstract

Physics-Informed Neural Networks (PINNs) have emerged as an iconic machine learning approach for solving Partial Differential Equations (PDEs). Although its variants have achieved significant progress, the empirical success of utilising feature mapping from the wider Implicit Neural Representations studies has been substantially neglected. We investigate the training dynamics of PINNs with a feature mapping layer via the limiting Conjugate Kernel and Neural Tangent Kernel, which sheds light on the convergence and generalisation of the model. We also show the inadequacy of commonly used Fourier-based feature mapping in some scenarios and propose the conditional positive definite Radial Basis Function as a better alternative. The empirical results reveal the efficacy of our method in diverse forward and inverse problem sets. This simple technique can be easily implemented in coordinate input networks and benefits the broad PINNs research.

Repo usage

Requirements

conda create -n RBF-PINN python=3.8
conda activate RBF-PINN
pip install [following...]

• torch == 2.0
• torchsummary
• numpy == 1.24.3
• matplotlib
• pyDOE

Citation

@misc{zeng2024training,
      title={Training dynamics in Physics-Informed Neural Networks with feature mapping}, 
      author={Chengxi Zeng and Tilo Burghardt and Alberto M Gambaruto},
      year={2024},
      eprint={2402.06955},
      archivePrefix={arXiv},
      primaryClass={cs.LG}
}

@article{Zeng2024RBFPINNNP,
  title={RBF-PINN: Non-Fourier Positional Embedding in Physics-Informed Neural Networks},
  author={Chengxi Zeng and Tilo Burghardt and Alberto M. Gambaruto},
  journal={ArXiv},
  year={2024},
  volume={abs/2402.08367},
}