  In this paper, the training dynamics of PINNs with a feature mapping layer
via the limiting Conjugate Kernel and Neural Tangent Kernel is investigated,
shedding light on the convergence of PINNs; Although the commonly used
Fourier-based feature mapping has achieved great success, we show its
inadequacy in some physics scenarios. Via these two scopes, we propose
conditionally positive definite Radial Basis Function as a better alternative.
Lastly, we explore the feature mapping numerically in a wide neural networks.
Our empirical results reveal the efficacy of our method in diverse forward and
inverse problem sets. Composing feature functions is found to be a practical
way to address the expressivity and generalisability trade-off, viz., tuning
the bandwidth of the kernels and the surjectivity of the feature mapping
function. This simple technique can be implemented for coordinate inputs and
benefits the broader PINNs research.
