  Neural networks (NNs) and linear stochastic estimation (LSE) have widely been
utilized as powerful tools for fluid-flow regressions. We investigate
fundamental differences between them considering two canonical fluid-flow
problems: 1. the estimation of high-order proper orthogonal decomposition
coefficients from low-order their counterparts for a flow around a
two-dimensional cylinder, and 2. the state estimation from wall characteristics
in a turbulent channel flow. In the first problem, we compare the performance
of LSE to that of a multi-layer perceptron (MLP). With the channel flow
example, we capitalize on a convolutional neural network (CNN) as a nonlinear
model which can handle high-dimensional fluid flows. For both cases, the
nonlinear NNs outperform the linear methods thanks to nonlinear activation
functions. We also perform error-curve analyses regarding the estimation error
and the response of weights inside models. Our analysis visualizes the
robustness against noisy perturbation on the error-curve domain while revealing
the fundamental difference of the covered tools for fluid-flow regressions.
