ResearchBenchmarksInverse-problem estimation on Euler equations No noiseInverse-problem estimation on Euler equations No noiseFollow0.9996Estimated kPI-HPQNN0.9950240.9962120.99740.998588Jun 17, 2026Evaluation resultsMethodMethodLinksEstimated kDensity Relative ErrorVelocity Relative ErrorPressure Relative ErrorPI-HPQNN2026.060.99961.351.111.19FPINN2026.060.99846.381.993.29HQNN2026.060.99733.699.111.24PINN2026.060.99523.156.435.53