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

In recent years, data-driven methods have been utilized to learn dynamical systems and partial differential equations (PDE). However, 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 PDEs via sparse regressions, such as the least absolute shrinkage and selection operator (Lasso) and sequential threshold ridge regression (STRidge). In this method, derivatives are calculated by automatic differentiation from the deep neural network, and equation form and coefficients are obtained with sparse regressions. The DL-PDE is tested with physical processes, governed by groundwater flow equation, contaminant transport equation, Burgers equation and Korteweg-de Vries (KdV) equation, for proof-of-concept and applications in real-world engineering settings. The proposed DL-PDE achieves satisfactory results when data are discrete and noisy.