Overview of Physics-Informed Neural Networks Applications

Abstract

When simulating various physical phenomena, the law of the phenomenon is often known in advance, in the form of a partial differential equation, that needs to be solved. Numerical methods, such as the finite element method, have been developed over decades, and these methods approximate the solution to the partial differential equation. However, these methods can be computationally demanding. On the oth- er hand, neural networks, can provide predictions that approximate the given partial differential equation. Neural networks are computationally more efficient than numerical methods, but they often face issues of generalization and consequently problems with solution accuracy. Insufficient generalization, among other things, can result from data collected from numerical simulations. In the last few years, physics-informed neural networks are being developed, for which it’s not necessary to gather data from simulations. These networks use automatic differentiation and during training, they minimize the residuals of the partial differ- ential equation, its initial, and boundary conditions. After training, these neural networks can be used as a replacement for traditional numerical solvers.