  The Neural Tangent Kernel (NTK) viewpoint is widely employed to analyze the
training dynamics of overparameterized Physics-Informed Neural Networks
(PINNs). However, unlike the case of linear Partial Differential Equations
(PDEs), we show how the NTK perspective falls short in the nonlinear scenario.
Specifically, we establish that the NTK yields a random matrix at
initialization that is not constant during training, contrary to conventional
belief. Another significant difference from the linear regime is that, even in
the idealistic infinite-width limit, the Hessian does not vanish and hence it
cannot be disregarded during training. This motivates the adoption of
second-order optimization methods. We explore the convergence guarantees of
such methods in both linear and nonlinear cases, addressing challenges such as
spectral bias and slow convergence. Every theoretical result is supported by
numerical examples with both linear and nonlinear PDEs, and we highlight the
benefits of second-order methods in benchmark test cases.
