The Neural Tangent Kernel (NTK) viewpoint represents a valuable approach to
examine the training dynamics of Physics-Informed Neural Networks (PINNs) in
the infinite width limit. We leverage this perspective and focus on the case of
nonlinear Partial Differential Equations (PDEs) solved by PINNs. We provide
theoretical results on the different behaviors of the NTK depending on the
linearity of the differential operator. Moreover, inspired by our theoretical
results, we emphasize the advantage of employing second-order methods for
training PINNs. Additionally, we explore the convergence capabilities of
second-order methods and address the challenges of spectral bias and slow
convergence. Every theoretical result is supported by numerical examples with
both linear and nonlinear PDEs, and we validate our training method on
benchmark test cases.

通过理论结果及数值示例，我们揭示部分微分方程的神经切向核心观点下，非线性微分算子呈现不同行为，以及使用二阶方法训练物理信息神经网络的优势和收敛能力，同时解决谱偏差和收敛速度慢的挑战。