Physics-Informed Neural Networks (PINN) are neural networks (NNs) that encode
model equations, like Partial Differential Equations (PDE), as a component of
the neural network itself. PINNs are nowadays used to solve PDEs, fractional
equations, integral-differential equations, and stochastic PDEs. This novel
methodology has arisen as a multi-task learning framework in which a NN must
fit observed data while reducing a PDE residual. This article provides a
comprehensive review of the literature on PINNs: while the primary goal of the
study was to characterize these networks and their related advantages and
disadvantages. The review also attempts to incorporate publications on a
broader range of collocation-based physics informed neural networks, which
stars form the vanilla PINN, as well as many other variants, such as
physics-constrained neural networks (PCNN), variational hp-VPINN, and
conservative PINN (CPINN). The study indicates that most research has focused
on customizing the PINN through different activation functions, gradient
optimization techniques, neural network structures, and loss function
structures. Despite the wide range of applications for which PINNs have been
used, by demonstrating their ability to be more feasible in some contexts than
classical numerical techniques like Finite Element Method (FEM), advancements
are still possible, most notably theoretical issues that remain unresolved.

文章综述了物理学启发的神经网络（PINN）的文献，并介绍了其特点和优缺点。此外，研究还包括了使用 PINN 以及它的许多其他变体解决 PDE、分数方程、积分微分方程和随机 PDE 的广泛应用领域，以及它们的定制化方法，如不同的激活函数、梯度优化技术、神经网络结构和损失函数结构。虽然该方法被证明在某些情况下比有限元方法更可行，但它仍面临理论问题尚未解决。