Scientific Machine Learning through Physics-Informed Neural Networks: Where we are and What’s next. (arXiv:2201.05624v1 [cs.LG])

Physic-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, and integral-differential equations. 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 larger variety of issues,
including physics-constrained neural networks (PCNN), where the initial or
boundary conditions are directly embedded in the NN structure rather than in
the loss functions. 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.



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