RBF-PINN: Non-Fourier Positional Embedding in Physics-Informed Neural Networks

TL;DR

This paper introduces RBF-PINN, a novel approach that replaces Fourier-based feature mapping with Radial Basis Functions in Physics-Informed Neural Networks, improving their performance on PDE problems.

Contribution

The paper proposes using Radial Basis Functions as a non-Fourier feature mapping in PINNs, addressing limitations of Fourier features and enhancing empirical results.

Findings

Effective across various forward and inverse PDE problems

Seamless integration into coordinate-based neural networks

Improved performance over traditional Fourier-based methods

Abstract

While many recent Physics-Informed Neural Networks (PINNs) variants have had considerable success in solving Partial Differential Equations, the empirical benefits of feature mapping drawn from the broader Neural Representations research have been largely overlooked. We highlight the limitations of widely used Fourier-based feature mapping in certain situations and suggest the use of the conditionally positive definite Radial Basis Function. The empirical findings demonstrate the effectiveness of our approach across a variety of forward and inverse problem cases. Our method can be seamlessly integrated into coordinate-based input neural networks and contribute to the wider field of PINNs research.