COMPARING AND CONTRASTING LINEAR STOCHASTIC ESTIMATES WITH NEURAL NETWORKS TO IDENTIFY IMPORTANT DIFFERENCES FOW REGRESSIONS ARE CAUSED BY FLUID

Abstract

When it comes to fuid-fow regressions, neural 
networks and linear stochastic estimation (LSE) are 
extensively used as powerful techniques. There are 
two canonical fuid-fow problems: (1) the estimation 
of high-order proper orthogonal decomposition 
coefficients from low-order counterparts for a two
dimensional cylindrical fow, and (2) the state 
estimation from the wall characteristics in a turbulent 
channel fow. We investigate the fundamental 
differences between them in the context of these two 
problems. LSE is compared to a multi-layer 
perceptron in the first problem (MLP). A nonlinear 
model, the convolutional neural network (CNN), is 
used to handle high-dimensional fuidfows in the 
channel fow example. The nonlinear NNs outperform 
the linear approaches in both circumstances because 
of the nonlinear activation functions. The estimate 
error and the response of the weights in models are 
also examined using error-curve studies. 
Visualization of the error-curve resilience against 
noise while highlighting the essential difference 
between the covered tools for fuid-fow regressions is 
the goal of our investigation. In terms of data 
estimation, control, reduced-order modelling, and 
turbulence modelling, NNs have actually shown their 
significant potentials at the basic level3–5. For the 
time being, we must depend on linear theories, 
notwithstanding current enthusiasm for the usage of 
NNs, due to their generalizability and transparency.  
If we can learn from the interaction between linear 
theories and neural networks, we can design more 
sophisticated NN-based methods for fuidfow 
analysis. The connection between NNs and linear 
approaches is of special significance here. Low
dimensionalization was achieved using an 
autoencoder (AE) in Milano and Koumoutsakos9's9 
research. AE based on multi-layer perceptron (MLP) 
with linear activation functions is comparable to 
proper orthogonal decomposition (POD)10, 
according to Tey's research. Murata et al.11 studied 
the similarity between an AE based on a 
convolutional neural network (CNN) and a POD in 
more recent research. NNs' nonlinear activation 
functions were shown to have significant strength. 
Nair & Goza12 recently examined the MLP, Gappy 
POD, and linear stochastic estimation (LSE) for the 
POD coeffcient estimate from local sensor 
measurements of the laminar wake of a fat plate 
using local sensor data. LSE, MLP, and CNN are 
used to compare the abilities of linear methods and 
NNs by comparing two canonical fluid fow 
regression problems whose complexities are 
different; 1. estimation of high-order POD coeffcients 
from low-order counterparts of a fow around a 
cylinder13, and 2. state estimation from information 
on the wall in a turbulent channel fow14. By 
concentrating on the influence of biases in NNs, 
optimization approaches and resilience against noisy inputs, we seek crucial features for 
their estimates.