Derivation of Equation

[Mahnoo06]

have two questions about the paper “ When and Why PINNs fail to train: A Neural Tangent Kernel Perspective” https://arxiv.org/pdf/2007.14527.pdf .

1. At the end of section 4, equation 4.10 (containing the exponential ) is derived. I was wondering what formula was used for its derivation. Unfortunately, I do not understand how you get from eq.4.9 to eq.4.10. Any leads on that would be helpful.

2. As this paper proves that the NTK becomes deterministic as we increase the width of the hidden layer. This means that NTK does not change too much compared to the NTK at initialisation. However, in section 7, it is also shown that eigenvalues decrease during training ( Figure3(c) ) so doesn't that imply that NTK would change as well which contradicts NTK being deterministic, right?

[psteinb]

Oh, I stumbled upon this too. Would love to hear some insights on this by @paraklas and @sifanw094. Thanks in advance.