  This work is concerned with discovering the governing partial differential
equation (PDE) of a physical system. Existing methods have demonstrated the PDE
identification from finite observations but failed to maintain satisfying
results against noisy data, partly owing to suboptimal estimated derivatives
and found PDE coefficients. We address the issues by introducing a noise-aware
physics-informed machine learning (nPIML) framework to discover the governing
PDE from data following arbitrary distributions. We propose training a couple
of neural networks, namely solver and preselector, in a multi-task learning
paradigm, which yields important scores of basis candidates that constitute the
hidden physical constraint. After they are jointly trained, the solver network
estimates potential candidates, e.g., partial derivatives, for the sparse
regression algorithm to initially unveil the most likely parsimonious PDE,
decided according to the information criterion. We also propose the denoising
physics-informed neural networks (dPINNs), based on Discrete Fourier Transform
(DFT), to deliver a set of the optimal finetuned PDE coefficients respecting
the noise-reduced variables. The denoising PINNs are structured into forefront
projection networks and a PINN, by which the formerly learned solver
initializes. Our extensive experiments on five canonical PDEs affirm that the
proposed framework presents a robust and interpretable approach for PDE
discovery, applicable to a wide range of systems, possibly complicated by
noise.

Numerical Analysis;Artificial Intelligence;Machine Learning;Numerical Analysis;Computational Physics