Mathematics

Advection-dominated problems are commonly noticed in nature, in engineering systems, and in a wide range of industrial processes. For these problems, linear compression methods (proper orthogonal decomposition and reduced basis method) are not suitable, as the Kolmogorov n-width (KnW) decay is slow, which leads to inefficient and inaccurate reduced order models. To accelerate the KnW decay there are few recent pre-processing techniques, in that I will discuss Neural-network shift based pre-processing technique and Radon Cumulative Distribution Transform (RCDT). Neural-network shift based pre-processing technique, automatically detects the optimal non-linear transformation of the full order model (FOM) solution manifold by exploiting a deep-learning architecture. It consists of two neural networks, 1) ShiftNet, which finds the optimal shift for the FOM solution manifold, to accelerate the KnW decay, and 2) InterpNet, which learns the reference configuration and able to reconstruct the shape of reference configuration for each shifted centroid distribution. Radon Cumulative Distribution transform (RCDT) is a non-linear and invertible transformations that guaranties certain linear separation theorems, emerging from computer vision field, and its application in model reduction of travelling-wave problems, shows the fastening of the KnW decay in RCDT space. In this talk, I will discuss the usage of these pre-processing techniques to develop purely data-driven reduced order models for advection-dominated problems like travelling waves, isentropic convective vortex and two-phase flows.