Training dynamics in Physics-Informed Neural Networks with feature
  mapping
Chengxi Zeng Tilo Burghardt Alberto M Gambaruto
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.