The differential neural tangent kernel (DNTK) is a tool for analyzing the training dynamics of overparameterized physics informed neural networks. In this work, we investigate the symmetry of the infinite DNTK under an orthogonal group, for two layer and deep neural networks separately, based on the explicit expression of the DNTK and recursive formula, respectively. The results are for all activations with or without a bias term, and shed insight into the structure of the infinite width DNTK. Then we prove that the integral operators over the unit sphere induced by operators from the same group orbit share the spectrum, and their reproducing kernel Hilbert spaces differ only by an orthogonal transformation, and that the operator fixed DNTK exhibits zonal symmetry on the unit sphere for symmetric operators. Thus the eigenvalue decay rate of the associated integral operator can be derived for shallow networks with RePU activation and zero bias initialization.

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