Robust Regression with Highly Corrupted Data via Physics Informed Neural Networks

TL;DR

This paper introduces LAD-PINN and MAD-PINN, novel physics-informed neural network methods designed to accurately recover solutions and parameters of PDEs from highly corrupted data, outperforming traditional approaches.

Contribution

The paper proposes two new robust PINN variants, LAD-PINN and MAD-PINN, that effectively handle highly corrupted data for PDE solution reconstruction and parameter discovery.

Findings

LAD-PINN successfully reconstructs solutions with high outlier contamination.

MAD-PINN improves accuracy by screening out corrupted data before applying PINNs.

Methods demonstrate effectiveness on Poisson, wave, and Navier-Stokes equations.

Abstract

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.