Physics-Informed Neural Operators. (arXiv:2207.05748v1 [cs.LG])

Standard neural networks can approximate general nonlinear operators,
represented either explicitly by a combination of mathematical operators, e.g.,
in an advection-diffusion-reaction partial differential equation, or simply as
a black box, e.g., a system-of-systems. The first neural operator was the Deep
Operator Network (DeepONet), proposed in 2019 based on rigorous approximation
theory. Since then, a few other less general operators have been published,
e.g., based on graph neural networks or Fourier transforms. For black box
systems, training of neural operators is data-driven only but if the governing
equations are known they can be incorporated into the loss function during
training to develop physics-informed neural operators. Neural operators can be
used as surrogates in design problems, uncertainty quantification, autonomous
systems, and almost in any application requiring real-time inference. Moreover,
independently pre-trained DeepONets can be used as components of a complex
multi-physics system by coupling them together with relatively light training.
Here, we present a review of DeepONet, the Fourier neural operator, and the
graph neural operator, as well as appropriate extensions with feature
expansions, and highlight their usefulness in diverse applications in
computational mechanics, including porous media, fluid mechanics, and solid



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