Physics-informed neural networks (PINNs) offer a promising avenue for
tackling both forward and inverse problems in partial differential equations
(PDEs) by incorporating deep learning with fundamental physics principles.
Despite their remarkable empirical success, PINNs have garnered a reputation
for their notorious training challenges across a spectrum of PDEs. In this
work, we delve into the intricacies of PINN optimization from a neural
architecture perspective. Leveraging the Neural Tangent Kernel (NTK), our study
reveals that Gaussian activations surpass several alternate activations when it
comes to effectively training PINNs. Building on insights from numerical linear
algebra, we introduce a preconditioned neural architecture, showcasing how such
tailored architectures enhance the optimization process. Our theoretical
findings are substantiated through rigorous validation against established PDEs
within the scientific literature.

物理启发的神经网络（PINNs）通过将深度学习与基本物理原理相结合，为解决偏微分方程中的正向和反向问题提供了一种有前途的方法。本研究从神经网络架构的角度深入探讨了 PINN 优化的复杂性，利用神经切向核（NTK），揭示了高斯激活提供了比其他激活函数更有效训练 PINNs 的优势。在数值线性代数的启示下，我们引入了一种经过预处理的神经网络架构，展示了这种定制架构如何增强优化过程。我们通过对科学文献中已有的偏微分方程进行严格验证，证实了我们的理论发现。