  Deep operator learning has emerged as a promising tool for reduced-order
modelling and PDE model discovery. Leveraging the expressive power of deep
neural networks, especially in high dimensions, such methods learn the mapping
between functional state variables. While proposed methods have assumed noise
only in the dependent variables, experimental and numerical data for operator
learning typically exhibit noise in the independent variables as well, since
both variables represent signals that are subject to measurement error. In
regression on scalar data, failure to account for noisy independent variables
can lead to biased parameter estimates. With noisy independent variables,
linear models fitted via ordinary least squares (OLS) will show attenuation
bias, wherein the slope will be underestimated. In this work, we derive an
analogue of attenuation bias for linear operator regression with white noise in
both the independent and dependent variables. In the nonlinear setting, we
computationally demonstrate underprediction of the action of the Burgers
operator in the presence of noise in the independent variable. We propose
error-in-variables (EiV) models for two operator regression methods,
MOR-Physics and DeepONet, and demonstrate that these new models reduce bias in
the presence of noisy independent variables for a variety of operator learning
problems. Considering the Burgers operator in 1D and 2D, we demonstrate that
EiV operator learning robustly recovers operators in high-noise regimes that
defeat OLS operator learning. We also introduce an EiV model for time-evolving
PDE discovery and show that OLS and EiV perform similarly in learning the
Kuramoto-Sivashinsky evolution operator from corrupted data, suggesting that
the effect of bias in OLS operator learning depends on the regularity of the
target operator.
