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

TL;DR

This paper introduces a noise-aware physics-informed machine learning framework that robustly discovers PDEs from noisy data using neural networks and denoising techniques, outperforming existing methods.

Contribution

It proposes a novel multi-task neural network approach combined with DFT-based denoising to improve PDE discovery accuracy under noisy conditions.

Findings

Robust PDE discovery demonstrated on five canonical PDEs.

The framework outperforms existing methods in noisy data scenarios.

The approach provides interpretable and accurate PDE coefficients.

Abstract

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.