Modelling process dynamics by the discovery of partial differential equations using a data-driven and hybrid modelling approach

The abundance of data and advances in data acquisition technologies have made data-driven approaches attractive to solve a multitude of problems. Differential equations deliver underlying models for most physical processes. Obtaining the fundamental physics underlying any data in the form of partial differential equations (PDEs) will facilitate modelling and prediction for systems where first principles modelling might not be feasible. Handling high dimensional spatiotemporal data holds high priority as it forms the underlying basis of many canonical models. Data-driven discovery has been addressed in the literature using Gaussian processes, artificial neural networks, and more recently, sparse regression techniques. Sparse optimization methods are used in multiple domains such as compressive sensing, scientific computing and to learn important features from data sets as they promote parsimony. Although there are multiple works done to discover PDEs using the sparse regression approach, there is no study about the optimal methods that can be utilized for a multitude of systems. We have carried out a detailed study of the sparse regression framework by inferring the best gradient estimation method and the optimal sparsity regularization method for different noise levels. These inferences have provided knowledge about handling uncertainties in the sparse regression framework. We have utilized these inferences and extended the work to discover a system of PDEs using data-driven and hybrid modelling approaches. The hybrid modelling approach was implemented to utilize the process knowledge to discover the system of PDEs. Through this framework, any partial information about the process can be incorporated into the PDE discovery framework in the form of mathematical constraints. The hybrid modelling approach improves the model accuracy due to the prior physical knowledge incorporated and also reduces the computational cost. Petroleum reservoirs are large scale distributed parameter processes from a systems and control theoretic perspective. We have developed an algorithm to discover parametric PDEs to explain the temperature dynamics of a steam-assisted gravity drainage(SAGD) process in an oil reservoir. The data required for the discovery was collected from a first-principles based commercial reservoir simulator and geomechanical simulator CMG-STARS sequentially coupled with FLAC3D. An ensemble of multiple realizations of temperature and permeability was generated which spanned across the spatial domain of the reservoir. A hybrid model was developed by incorporating the permeability values in the discovery framework and has higher accuracy compared to the data-driven model. PDEs were discovered for each realization and then integrated to form a spatially varying parametric PDE explaining the temperature dynamics in an oil reservoir.