DL-PDE: Deep-learning based data-driven discovery of partial differential equations from discrete and noisy data

TL;DR

DL-PDE is a deep learning approach that combines neural networks and sparse regression to discover PDEs from noisy, limited data, effectively identifying underlying physical laws in complex systems.

Contribution

The paper introduces DL-PDE, a novel method that integrates deep learning with sparse regression for PDE discovery from noisy and limited data.

Findings

Successfully discovers PDEs from noisy data

Performs well with limited data samples

Validated on multiple physical systems

Abstract

In recent years, data-driven methods have been developed to learn dynamical systems and partial differential equations (PDE). The goal of such work is discovering unknown physics and the corresponding equations. However, prior to achieving this goal, major challenges remain to be resolved, including learning PDE under noisy data and limited discrete data. To overcome these challenges, in this work, a deep-learning based data-driven method, called DL-PDE, is developed to discover the governing PDEs of underlying physical processes. The DL-PDE method combines deep learning via neural networks and data-driven discovery of PDE via sparse regressions. In the DL-PDE, a neural network is first trained, and then a large amount of meta-data is generated, and the required derivatives are calculated by automatic differentiation. Finally, the form of PDE is discovered by sparse regression. The proposed method is tested with physical processes, governed by groundwater flow equation, convection-diffusion equation, Burgers equation and Korteweg-de Vries (KdV) equation, for proof-of-concept and applications in real-world engineering settings. The proposed method achieves satisfactory results when data are noisy and limited.