One of the popular method used for Model Order Reduction (MOR) of non-linear Partial Differential Equations (PDEs) is the Proper Orthogonal Decomposition (POD) with Galerkin Projection (GP). The POD – GP method finds the optimal subspace in L2 sense that encapsulates the system’s dynamics and projects the governing PDE in this subspace. This method finds utility in large classes of problems. However, advection dominated PDEs require a large number of dimensions, or modes, to encapsulate the system dynamics properly. Therefore, the POD – GP method is observed to fail for advection dominated PDEs due to truncation of the higher modes. Often the alternative approach used for advection dominated PDEs are non-linear MOR, which are frequently based on Deep Learning methods and lack explainability. In this thesis work, we formulate a method that combines a coupled linear and non-linear MOR, such that it overcomes the issue of POD – GP while still retaining the formulation. In the proposed POD – CAE projection method, the first few POD modes are used for GP while the truncated modes are projected using CAE and are progressed using an LSTM model. We show that this formulation provides the linear MOR with necessary information about the dynamics of the truncated modes through coupling between the two models. We apply the POD – CAE formulation to solve the Burgers’ equation and radiative heat equation to demonstrate the capability, flexibility, and explainability of the proposed formulation. Our results show that the formulation is able to tackle the challenges faced by the POD – GP method. Additionally, examining the low dimensional space allows the user to isolate the sources of error in the method, thereby providing explainability.

Keywords: Non-Linear Model Order Reduction, Proper Orthogonal Decomposition, POD — CAE.