POD-Galerkin reduced order models and physics-informed neural networks for solving inverse problems for the Navier–Stokes equations

Advanced Modeling and Simulation in Engineering Sciences

Table 1 The PINNs results for the data mean squared error and the mean squared errors corresponding to the residual functions defining the POD-Galerkin ROM DAE

Initial \(a_l\)	\({\textbf {E}}_{\text {data}}({\varvec{w}})\)	\({\textbf {E}}_\textbf{1}({\varvec{w}})\)	\({\textbf {E}}_\textbf{2}({\varvec{w}})\)	PINN \(a_l\)
0	\(9.152303* 10^{-6}\)	0.00045994046	\(3.0014705* 10^{-6}\)	2.7291148
5	\(9.2302425* 10^{-6}\)	0.0009832102	\(3.0044891* 10^{-6}\)	2.7288582
10	0.00020264629	0.00036979085	\(2.9998373* 10^{-6}\)	2.7270014
20	\(1.4284992* 10^{-5}\)	0.0016940477	\(3.0059625* 10^{-6}\)	2.7295387

The values of the errors are reported for different initial values of the added weight \(a_l\), the PINN identified value of \(a_l\) is reported in the last column. The PINNs are run for 30000 epochs with a learning rate of \(1 * 10^{-3}\)