Skip to content
DSML.KZ
Publications

Article · Mathematics · October 2, 2026

Identifiability and Conditioning of Flux-Parameter Inversion for the Buckley–Leverett Equation

Samson Dawit Bekele, Yerzhan Kenzhebek, Saltanbek Talapedenovich MUKHAMBETZHANOV, Saida Tastanova, Dagmawi Lemma, Timur Imankulov

Samson Dawit Bekele
Open via DOI

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.

Model Reduction and Neural NetworksReservoir Engineering and Simulation MethodsNumerical methods in inverse problemsBuckley-Leverett equationstructural identifiabilitycondition number

More by Samson Dawit Bekele