Figure 3
From: ML for fast assimilation of wall-pressure measurements from hypersonic flow over a cone
Pressure spectra at the four sensors \(s_1-s_4\); measurements \(\varvec{m}\) corresponding to different solutions \(\varvec{c}^{\star }\) of the data assimilation problem. Circles: experimental data (99% confidence interval is \(\log _{10}\left( |\hat{p}|\right) \pm 0.03\)). Grey line: output \(\varvec{m}\) corresponding to \(\varvec{c}_{\scriptscriptstyle {\widetilde{\mathcal {N}}}}^{\star }\), explicit solution based on the linearized Navier–Stokes operator \(\widetilde{\mathcal {N}}\). Blue line with dots: output \(\mathcal {M}_{\scriptscriptstyle {S}}(\varvec{c}_{\scriptscriptstyle {S}}^{\star })\), EnVar solution (reference based solely on direct numerical simulations). Red solid line: output \(\mathcal {M}_{\scriptscriptstyle {D}}(\varvec{c}_{\scriptscriptstyle {D}}^{\star })\), FastDA solution (proposed method). Red transparent band: ± three times the validation error of the DeepONet, \(\mathcal {M}_{\scriptscriptstyle {D}}(\varvec{c}_{\scriptscriptstyle {D}}^{\star }) \pm 3\overline{\varepsilon }_{\scriptscriptstyle {D}}\). Black dashed line: output \(\mathcal {M}_{\scriptscriptstyle {S}}(\varvec{c}_{\scriptscriptstyle {D}}^{\star })\), FastDA solution simulated with the Navier–Stokes operator \(\mathcal {N}\).
