A hybrid modelling approach to model process dynamics by the discovery of a system of partial differential equations

Modelling temperature dynamics of the SAGD process in an oil reservoir by the discovery of parametric partial differential equations

“…Subscripts m and n represent the number of temporal and spatial points, respectively. Utilizing the results from the gradient estimation method comparison for PDE discovery in the literature, [12] polynomial interpolation is utilized to estimate the gradient values for the terms shown in the candidate library. We perform least squares estimation combined with sequentially threshold ridge regression [5] to obtain parsimonious PDEs.…”

“…We discovered the temperature dynamics using two different approaches and compared the models obtained. a. Data-driven modelling: Although hybrid models proved to generate better PDE models in the literature, [12] developing data-driven models is vital as data availability for the petrophysical parameter might not be available in some cases. We developed two different data-driven models with modifications in the candidate library based on information obtained from the 1D heat equation.…”

“…Tulleken proposed identifying a linearly parameterized dynamic regression model using Bayesian techniques [10] ; non-linear system identification utilizing neural networks considering partial physical knowledge was proposed by Xiong and Jutan. [11] A hybrid modelling approach to discover a system of PDEs by assuming partial information about the process was proposed by Raviprakash et al [12] The hybrid modelling framework is extensively applied in process industries and in chemical engineering practices. [13][14][15] The above-mentioned studies corresponding to the discovery of differential equations consider a single PDE or a system of PDEs with constant coefficients.…”