Solving parametric partial differential equations using artificial intelligence is taking the pace. It is primarily because conventional numerical solvers are computationally expensive and require significant time to converge a solution. However, physics informed deep learning as an alternate learns functional spaces directly and provides approximation reasonably fast compared to conventional numerical solvers. The Fourier transform approach directly learns the generalized functional space using deep learning among various approaches. This work proposes a novel deep Fourier neural network that employs a Fourier neural operator as a fundamental building block and employs spectral feature aggregation to extrude the extended information. The proposed model offers superior accuracy and lower relative error. We employ one and two-dimensional time-independent as well as two-dimensional time-dependent equations. We employ three benchmark datasets to evaluate our contributions, i.e., Burgers' equation as one dimensional, Darcy Flow equation as two dimensional, and Navier-Stokes as two spatial dimensional with one temporal dimension as benchmark datasets. We further employ a case study of fluid-structure interaction used for the machine component designing process.We employ a computation fluid dynamics simulation dataset generated using the ANSYS-CFX software system to evaluate the regression of the temporal behavior of the fluid. Our proposed method achieves superior performance on all four datasets employed and shows improvements to baseline. We achieve a reduced relative error on the Burgers' equation by approximately 30%, Darcy Flow equation by approximately 35%, and Navier-Stokes equation by approximately 20%.
Partial differential equation approximation, physics informed neural network, computational fluid dynamics, neural operators, functional spaces, spectral feature learning.
@ARTICLE{Rafiq2022a,
author={Rafiq, Muhammad and Rafiq, Ghazala and Jung, Ho-Youl and Choi, Gyu Sang},
journal={IEEE Access},
title={SSNO: Spatio-Spectral Neural Operator for Functional Space Learning of Partial Differential Equations},
year={2022},
volume={10},
number={},
pages={15084-15095},
doi={10.1109/ACCESS.2022.3148401}
}
@ARTICLE{Rafiq2022b, author={Rafiq, Muhammad and Rafiq, Ghazala and Choi, Gyu Sang}, journal={IEEE Access}, title={DSFA-PINN: Deep Spectral Feature Aggregation Physics Informed Neural Network}, year={2022}, volume={}, number={}, pages={1-1}, doi={10.1109/ACCESS.2022.3153056}}