RBF-PINN: Non-Fourier Positional Embedding in Physics-Informed Neural Networks
Chengxi Zeng, Tilo Burghardt, Alberto M Gambaruto
TL;DR
The paper addresses the limitations of Fourier-based feature mappings in physics-informed neural networks (PINNs), highlighting artifacts and poor scaling in high dimensions. It proposes a conditional positive definite Radial Basis Function (RBF) feature mapping, specifically a normalized Gaussian RBF with trainable centers and a polynomial augmentation, to improve representational power and training stability. Across forward and inverse PDE tasks (diffusion, heat, Poisson, Burgers, Navier–Stokes), the RBF-PINN demonstrates substantial accuracy gains, robustness to noise, and better handling of multiscale phenomena compared to Fourier encodings. The method integrates smoothly with existing PINN frameworks and offers potential benefits for other coordinate-based neural representations beyond PDE contexts.
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.