ResearchBenchmarksInverse-problem estimation on Euler equations 5% noiseFollow1.0212Estimated k ValuePINN0.9834480.9932491.003051.012851Jun 17, 2026Evaluation ResultsMethodMethodLinksEstimated k ValueDensity (rho) Relative ErrorVelocity (u) Relative ErrorPressure (p) Relative ErrorPINN2026.061.02122.64.915.15PI-HPQNN2026.061.00138.562.842.45FPINN2026.060.99851.451.411.77HQNN2026.060.98492.527.026.96