Identification of nonlinear conservation laws for multiphase flow based on Bayesian inversion

• Published in: Nonlinear dynamics Vol. 111; no. 19; pp. 18163 - 18190
• Main Authors: Evje, Steinar, Skadsem, Hans Joakim, Nævdal, Geir
• Format: Journal Article
• Language: English
• Published: Dordrecht Springer Netherlands 01.10.2023; Springer Nature B.V
• Subjects: Algorithms; Automotive Engineering; Bayesian analysis; Classical Mechanics; Conservation laws; Control; Displacement; Dynamical Systems; Engineering; Gaussian process; Identification; Identification methods; Inverse problems; Iterative methods; Kalman filters; Mechanical Engineering; Multiphase flow; Neural networks; Nonlinear dynamics; Original Paper; Partial differential equations; Random processes; Smoothness; Vibration; Data-driven mathematical modelling; Multiphase model; Bayesian inversion; Nonlinear conservation law; Gaussian random process; Iterative ensemble (Kalman filter) smoother
• ISSN: 0924-090X; 1573-269X
• DOI: 10.1007/s11071-023-08817-9

Conservation laws of the generic form c t + f ( c ) x = 0 play a central role in the mathematical description of various engineering related processes. Identification of an unknown flux function f ( c ) from observation data in space and time is challenging due to the fact that the solution c ( x ,  t ) develops discontinuities in finite time. We explore a Bayesian type of method based on representing the unknown flux f ( c ) as a Gaussian random process (parameter vector) combined with an iterative ensemble Kalman filter (EnKF) approach to learn the unknown, nonlinear flux function. As a testing ground, we consider displacement of two fluids in a vertical domain where the nonlinear dynamics is a result of a competition between gravity and viscous forces. This process is described by a multidimensional Navier–Stokes model. Subject to appropriate scaling and simplification constraints, a 1D nonlinear scalar conservation law c t + f ( c ) x = 0 can be derived with an explicit expression for f ( c ) for the volume fraction c ( x ,  t ). We consider small (noisy) observation data sets in terms of time series extracted at a few fixed positions in space. The proposed identification method is explored for a range of different displacement conditions ranging from pure concave to highly non-convex f ( c ). No a priori information about the sought flux function is given except a sound choice of smoothness for the a priori flux ensemble. It is demonstrated that the method possesses a strong ability to identify the unknown flux function. The role played by the choice of initial data c 0 ( x ) as well various types of observation data is highlighted.