Simulations of calibrated models often provide the only way to predict the nonlinear flow in geological formations. Widely used methods of machine learning (ML) in the scientific community appear to be well-suited for solving such nonlinear problems. However, classical ML methods require a large amount of data, both for model parameters and model response. Unfortunately, many applications in the geosciences can only provide a limited number of datasets. This data sparsity results from both a limited number of available measurement data and high computational effort required for the numerical simulation of realistic models. Multiphase flow of CO2 in deep geological formations is an undisputed representative of this class of problems.
In the current project, we intend to develop an ML method that will be capable of adaptively approximating the local nonlinearity of the physical problem even in the case of only a small amount of available data. The project aims to leverage the relationship between Bayesian inference and information theory in its goal-oriented manifestation to adaptively localize the nonlinearity of the physical problem using observational data and simulation results. Following the current trend in stochastic model reduction, we train a mathematically optimal response surface using a limited amount of information from the original CO2 model concerning observational data.
The novelty of the current project lies in the extension of the arbitrary Multi-Resolution Polynomial Chaos framework (currently developed by the applicants) into adaptive and sparse reconstruction based on Bayesian theory followed by information-theoretic arguments. An optimal method for a sparse structure of the piecewise polynomial representation is derived to maximize the quality of Bayesian parameter inference at the given computational cost. Bayesian model evidence provides the necessary mathematical methods. Following the idea of Bayesian experimental design, we will maximize the expected utility and identify the optimal set of parameters. The application of information-theoretic arguments will help localize the parameter range of highest relevance.
The combination of Bayesian inference and information-theoretic arguments leads to an iterative and adaptive improvement of the response surface. This new approximation enables the calibration of highly nonlinear models while significantly reducing computational effort, with the quantified post-calibration uncertainty focusing its approximation quality on the parameter range of highest interest.
