Machine Learning of Partial Differential Equations from Noise Data

TL;DR

This paper introduces a frequency domain method using Fourier transforms to accurately identify partial differential equations from noisy data, overcoming noise-related challenges in traditional approaches.

Contribution

It proposes a novel frequency domain identification technique combined with a sparse criterion for robust PDE discovery from low SNR data.

Findings

High accuracy in identifying PDE structures from noisy data

Robustness demonstrated across various canonical equations

Effective noise elimination using low frequency components

Abstract

Machine learning of partial differential equations from data is a potential breakthrough to solve the lack of physical equations in complex dynamic systems, but because numerical differentiation is ill-posed to noise data, noise has become the biggest obstacle in the application of partial differential equation identification method. To overcome this problem, we propose Frequency Domain Identification method based on Fourier transforms, which effectively eliminates the influence of noise by using the low frequency component of frequency domain data to identify partial differential equations in frequency domain. We also propose a new sparse identification criterion, which can accurately identify the terms in the equation from low signal-to-noise ratio data. Through identifying a variety of canonical equations spanning a number of scientific domains, the proposed method is proved to have high accuracy and robustness for equation structure and parameters identification for low signal-to-noise ratio data. The method provides a promising technique to discover potential partial differential equations from noisy experimental data.