DL-PDE: Deep-Learning Based Data-Driven Discovery of Partial Differential Equations from Discrete and Noisy Data (Communications in Computational Physics, 2021)
In recent years, data-driven methods have been developed to learn dynamical systems and partial differential equations (PDE). The goal of such work is to discover unknown physics and corresponding equations. However, prior to achievingthis goal, major challenges remain to be resolved, including learning PDE under noisydata 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 deeplearning via neural networks and data-driven discovery of PDE via sparse regressions.In the DL-PDE, a neural network is first trained, then a large amount of meta-data isgenerated, and the required derivatives are calculated by automatic differentiation. Finally, the form of PDE is discovered by sparse regression. The proposed method istested with physical processes, governed by the diffusion equation, the convection-diffusion equation, the Burgers equation, and the Korteweg-de Vries (KdV) equation,for proof-of-concept and applications in real-world engineering settings. The proposedmethod achieves satisfactory results when data are noisy and limited.