Identifiability and Conditioning of Flux-Parameter Inversion for the Buckley–Leverett Equation
- 1Department of Computer Science, Al-Farabi Kazakh National University, 71 Al-Farabi Avenue, Almaty 050040, Kazakhstan
- 2Department of Television and Media Technologies, Tashkent University of Information Technologies Named After Muhammad Al-Khwarizmi, 108 Amir Temur Avenue, Tashkent 100200, Uzbekistan
- 3Department of Computer Science, Addis Ababa University, Addis Ababa P.O. Box 1176, Ethiopia
Abstract
We analyse inversion of the viscosity ratio M in the Buckley–Leverett equation, distinguishing structural identifiability, observation-operator conditioning, and numerical estimation error. The Riemann solution gives M=4s2−4s from the shock speed s, with relative condition number κ(M)=21+M/(1+M−1), which diverges as M↓0. The continuum physics objective admits exact solution–parameter pairs for every M>0, so identifying the true ratio requires observations. We examine numerical recovery using a viscous physics-informed neural network (PINN) with positive parameterisation and 30 paired repetitions per setting. At M=3, the median parameter error decreases from 514.25% without observations to 6.13% with 25 noisy observations. Additional data decrease the median field error, while median parameter errors are nonmonotone. Over M∈[0.25,10], κ decreases from 18.94 to 2.86, whereas the smooth-branch Cramér–Rao relative scale for 50 scattered observations with 5% Gaussian noise levels off near 7%. Shock crossings supply additional nonregular information. Finite-network physics losses remain small across the tested ratios, with different orderings on training and held-out points. Scalar controls assess finite-resolution viscous-model discrepancy; complementary PINN tests show poorer recovery as the front sharpens. Recovery accuracy therefore depends jointly on the observation operator, regularisation, and numerical estimator.
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BibTeX
@article{bekele2026identifiability,
title = {{Identifiability and Conditioning of Flux-Parameter Inversion for the Buckley–Leverett Equation}},
author = {Samson Dawit Bekele and Yerzhan Kenzhebek and Saltanbek Mukhambetzhanov and Saida Tastanova and Dagmawi Lemma and Timur Imankulov},
journal = {Mathematics},
year = {2026},
month = oct,
doi = {10.3390/math14193575},
url = {https://www.mdpi.com/2227-7390/14/19/3575},
note = {CC BY 4.0},
}
RIS
TY - JOUR
TI - Identifiability and Conditioning of Flux-Parameter Inversion for the Buckley–Leverett Equation
AU - Samson Dawit Bekele
AU - Yerzhan Kenzhebek
AU - Saltanbek Mukhambetzhanov
AU - Saida Tastanova
AU - Dagmawi Lemma
AU - Timur Imankulov
PY - 2026
DA - 2026/10/02
JO - Mathematics
DO - 10.3390/math14193575
UR - https://www.mdpi.com/2227-7390/14/19/3575
AB - We analyse inversion of the viscosity ratio M in the Buckley–Leverett equation, distinguishing structural identifiability, observation-operator conditioning, and numerical estimation error. The Riemann solution gives M=4s2−4s from the shock speed s, with relative condition number κ(M)=21+M/(1+M−1), which diverges as M↓0. The continuum physics objective admits exact solution–parameter pairs for every M>0, so identifying the true ratio requires observations. We examine numerical recovery using a viscous physics-informed neural network (PINN) with positive parameterisation and 30 paired repetitions per setting. At M=3, the median parameter error decreases from 514.25% without observations to 6.13% with 25 noisy observations. Additional data decrease the median field error, while median parameter errors are nonmonotone. Over M∈[0.25,10], κ decreases from 18.94 to 2.86, whereas the smooth-branch Cramér–Rao relative scale for 50 scattered observations with 5% Gaussian noise levels off near 7%. Shock crossings supply additional nonregular information. Finite-network physics losses remain small across the tested ratios, with different orderings on training and held-out points. Scalar controls assess finite-resolution viscous-model discrepancy; complementary PINN tests show poorer recovery as the front sharpens. Recovery accuracy therefore depends jointly on the observation operator, regularisation, and numerical estimator.
SN - 2227-7390
ER -