Deep-learning of parametric partial differential equations from sparse and noisy data

Data-driven methods have recently made great progress in the discovery of partial differential equations (PDEs) from spatial-temporal data. However, several challenges remain to be solved, including sparse noisy data, incomplete library, and spatially or temporally varying coefficients. In this work, a new framework, which combines neural network, genetic algorithm, and stepwise methods, is put forward to address all of these challenges simultaneously. In the framework, a trained neural network is utilized to calculate derivatives and generate a large amount of meta-data, which solves the problem of sparse noisy data. Next, the genetic algorithm is used to discover the form of PDEs and corresponding coefficients, which solves the problem of the incomplete initial library. Finally, a stepwise adjustment method is introduced to discover parametric PDEs with spatially or temporally varying coefficients. In this method, the structure of a parametric PDE is first discovered, and then the general form of varying coefficients is identified. The proposed algorithm is tested on the Burgers equation, the convection-diffusion equation, the wave equation, and the KdV equation. Results demonstrate that this method is robust to sparse and noisy data, and is able to discover parametric PDEs with an incomplete initial library.