| Abstract [eng] |
This thesis focuses on a problem of determining field distribution of a particular measure u throughout a highly heterogeneous and layered permeability domain assuming some driving function. An elliptic Partial Differential Equation, used to describe such distribution at an equilibrium state, is called Poisson's Equation which originates from Darcy's Law. The aforementioned equation describes diffusion and has no temporal dimension, thus requiring sufficient boundary conditions to be uniquely solved using numerical methods. After performing a comparison study of two such methods known as Finite Difference Method (FDM) and Finite Volume Method (FVM), a proposition of an alternative approach of using Artificial Neural Network (ANN) to solve a given problem was made. The ANN is referred to as a Coordinate Discretized Neural Network (CDNN) throughout the thesis. Both numerical methods were implemented on a two-dimensional grid and compared with each other on highly heterogeneous and layered permeability domains. Since FVM is is both locally and globally conservative method it outperformed FDM in several ways. The latter created non-monotonous solutions with sharp transitions in between the sub-domains and overall drew unrealistic maps of u that were excessively impacted by the underlying permeability domain. Conclusions were drawn from these observations to train CDNN solely on FVM's generated data, keeping in mind how a proper method should behave. CDNN produced accurate results, managing to replicate the FVM, even on some complex permeability distributions. However, high noise contamination present in the results leaves room for further network architecture improvement and there might be even possibilities to reformulate the modelling approach to improve the results. Suggestions for further development and possible network enhancements, were discussed in greater detail in the concluding discussion chapter. |
