Noise-aware Physics-informed Machine Learning for Robust PDE Discovery

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 performance 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. Our proposals are twofold. First, we propose a couple of neural networks, namely solver and preselector, which yield an interpretable neural representation of the hidden physical constraint. After they are jointly trained, the solver network approximates potential candidates, e.g., partial derivatives, which are then fed to the sparse regression algorithm that initially unveils the most likely parsimonious PDE, decided according to the information criterion. Second, we 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' structures are compartmentalized 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.