  Physics-informed neural networks (PINNs) have been proposed to solve two main
classes of problems: data-driven solutions and data-driven discovery of partial
differential equations. This task becomes prohibitive when such data is highly
corrupted due to the possible sensor mechanism failing. We propose the Least
Absolute Deviation based PINN (LAD-PINN) to reconstruct the solution and
recover unknown parameters in PDEs - even if spurious data or outliers corrupt
a large percentage of the observations. To further improve the accuracy of
recovering hidden physics, the two-stage Median Absolute Deviation based PINN
(MAD-PINN) is proposed, where LAD-PINN is employed as an outlier detector
followed by MAD screening out the highly corrupted data. Then the vanilla PINN
or its variants can be subsequently applied to exploit the remaining normal
data. Through several examples, including Poisson's equation, wave equation,
and steady or unsteady Navier-Stokes equations, we illustrate the
generalizability, accuracy and efficiency of the proposed algorithms for
recovering governing equations from noisy and highly corrupted measurement
data.
