Learning Reduced Order Mappings of Navier-Stokes

Abstract

Solving Partial Differential Equations (PDEs) in engineering such as Navier-Stokes is incredibly computationally expensive and complex. Without analytical solutions, numerical solutions can take ages to simulate at great expense. In order to reduce this cost, neural networks may be used to compute approximations of the solution for use during engineering processes. PCA-net is a neural network approach that reduces the dimensionality of the input and output data for PDEs in order to allow mapping from a high-dimensional input and output function with a fully connected neural network through the use of Principal Component Analysis (PCA). In this paper, PCA-net is applied to Navier-Stokes with varying viscosities to test the generalization of PCA-net on viscosity parameters. Training is done on four discrete viscosities, while testing is done on continuous viscosities, extrapolating and interpolating around the training set. Results shows good performance on low viscosities, both with interpolation and extrapolation. Mid-to-high viscosity interpolation shows lesser performance, with high viscosity extrapolation diverging to great error. Omitting high viscosities, performance over varying viscosities is close to that shown by previous research.