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 3 The computational time taken by different tasks needed for the implementation of the POD-Galerkin PINN ROM (i) the second column reports the time taken for performing the projection of the equations (computing the reduced quantities in the reduced DAE system) in parallel setting using 24 processors, (ii) the third column details the CPU time (also using 24 processors in parallel) consumed for the computation of the PINNs output data represented by the \(L^2\) projection coefficients of the different variables

\(\varvec{N_u=N_p=N_S}\)	\(\varvec{T_{\text {proj,DAE}}}\) in s	\(\varvec{T_{\text {proj,data}}}\) in s	\(\varvec{T_{\text {PINNs,T}}}\) in min	\(\varvec{T_{\text {PINN,F}}}\) in s	\(\varvec{\text {Speedup}}\)
10	8.269	37.6084	58.01	0.004612	\(2.4438 * 10^6\)
20	25.219	59.5178	59.94	0.004715	\(2.3711 * 10^6\)
30	55.0379	77.1995	61.78	0.004778	\(2.3557 * 10^6\)
40	96.2729	100.27687	58.01	0.004873	\(2.2907 * 10^6\)
50	145.838	126.1021	71.7	0.006929	\(1.6261 * 10^6\)
60	207.629	159.0223	70.63	0.006481	\(1.7393 * 10^6\)
70	289.609	187.9712	59.91	0.006036	\(1.8584 * 10^6\)
80	387.812	233.1214	72.06	0.005258	\(2.1340 * 10^6\)
90	462.827	270.7364	65.49	0.005462	\(2.0480 * 10^6\)
100	649.555	322.3108	89.91	0.005543	\(2.0314 * 10^6\)
120	850.28	428.9114	101.0	0.005463	\(2.0368 * 10^6\)
150	1289.12	625.9247	136.8	0.006089	\(1.8480 * 10^6\)

(iii) The fourth column shows the time required for running the PINNs on the Graphics Processing Unit (GPU), (iii) the fifth column indicates the time taken for carrying out the forward online computations of the PINNs for the approximation of the reduced variables. (iv) The last column reports the speedup achieved by the reduced order model. The speedup is calculated as follows \(\text {Speedup} = \frac{T_\text {Off}}{T_\text {PINN,F}+T_\text {Onl,Forces}}\), where \(T_\text {Off} = 11,460\,{\textrm{s}}\) is the CPU time needed for simulating the FOM for the snapshots time window using 24 processors in parallel, and \(T_\text {Onl,Forces}\) is the time consumed for assembling the reduced forces in the online stage which has taken around \(7-10 * 10^{-5} \,{\textrm{s}}\) in the current experiments. The table shows the computational costs for different sizes of reduced modes